Cuntz–Krieger algebras, minimal systems and projectionless algebras

Written by GPT-6.1 Sol (OpenAI) in Codex, Ultra setting, October 2026. Self-checked by the writing AI. Public domain (CC0).

A finite transition matrix can encode the K-theory of a C*-algebra. A minimal homeomorphism can instead produce an algebra whose only projections are zero and one, even though its K-groups remain large. We finish the course by computing both kinds of examples and comparing them with free-group algebras.

We reuse the PV sequence, AF continuity, the connected-space trace theorem, and the gauge-corner argument of Cuntz algebras. All tensor products below are spatial. Write \(\mathcal K\) for the compact operators and normalize circle Haar measure to have mass one.

1. Relations and the stationary core

Initially let \(A=(a_{ij})\) be an \(n\)-by-\(n\) zero–one matrix with no zero row or column. The universal unital algebra \(\mathcal O_A\) is generated by partial isometries \(s_i\). Put \(p_i=s_i s_i^*\) and impose the full relations

The last equation alone does not define the intended algebra. If every entry of \(A\) is one, (1.1) gives the Cuntz algebra \(\mathcal O_n\).

To see existence, let \(\Omega_A\) be the set of infinite sequences \(x_1x_2\cdots\) with \(a_{x_k x_{k+1}}=1\). On \(\ell^2(\Omega_A)\), define \(S_i\delta_x=\delta_{ix}\) when \(a_{i x_1}=1\), and zero otherwise. Its range consists of sequences beginning with \(i\); its initial projection selects allowed first letters. These operators satisfy (1.1), and every \(p_i\) is nonzero because every vertex begins an infinite path. The supremum norm over all representations of the relations is finite on polynomials, since every generator is a contraction. Quotienting by its zero seminorm and completing gives the universal algebra. The path representation establishes nonzero vertices; we do not assert that it is faithful.

Universality supplies a point-norm continuous gauge action \(\gamma_z(s_i)=zs_i\). Continuity follows first on polynomials and then by norm approximation. Its average \(E_\gamma\) is faithful: for positive \(x\), each state applied to \(\gamma_z(x)\) is a nonnegative continuous function; a zero average makes its value at \(z=1\) zero. States separate positive elements.

For a word \(\mu=i_1\cdots i_r\), write \(s_\mu=s_{i_1}\cdots s_{i_r}\). For each terminal vertex \(j\), retain words with consecutive allowed transitions and \(a_{i_rj}=1\). Set \(w_{\mu,j}=s_\mu p_j\); at length zero set \(w_{\varnothing,j}=p_j\). Cancellation using (1.1) gives, for words of the same length,

Indeed \(s_i^*s_l=0\) for distinct letters, while a matching letter contributes \(q_i\); on an allowed next vertex this projection acts as the identity. Induction cancels the entire word. The equalities \(p_j=s_j q_j s_j^*\) and \(q_j=\sum_k a_{jk}p_k\) also give the refinement

Thus at length \(r\) the elements \(w_{\mu,j}w_{\nu,j}^*\) form full matrix blocks, one for each terminal vertex. They are nonzero by (1.2). Their diagonal sum is one: this is true at length zero, and (1.3) preserves that sum. No zero columns ensure that each vertex has at least one prefix of every length. If \(N_{r,j}\) counts those prefixes, the resulting finite-dimensional algebra is

These blocks exhaust the fixed algebra. To check the density assertion, expand \(s_i=\sum_j a_{ij}s_i p_j\) and repeatedly cancel adjacent starred and unstarred letters. Products reduce to linear combinations of \(s_\mu p_j s_\nu^*\), with allowed terminal transitions, including empty words. Their degree is \(|\mu|-|\nu|\). Averaging a polynomial discards precisely the nonzero degrees; a degree-zero term belongs to its equal-length block. Averaging approximating polynomials proves \(F_A=\mathcal O_A^\gamma=\overline{\bigcup_r F_r}\).

Use the rank-one class in block \(j\) as coordinate \(e_j\). Formula (1.3) sends it to \(\sum_k a_{jk}e_k\). Consequently, with column vectors and \(T=A^t\),

where \(\mathbf1=(1,\ldots,1)^t\) and \([v,r]=[Tv,r+1]\). This uses finite-dimensional K-theory and continuity from Lesson 16.

Lemma 1.1 (aperiodic path compression). Suppose that the zero–one matrix \(A\) is irreducible and is not a permutation matrix. Given a vertex \(j\), finitely many normal words \(s_\alpha p_k s_\beta^*\) of nonzero degree, and any prescribed lower bound on length, there is an admissible word \(\eta\) beginning at \(j\) and an allowed next vertex \(l\) such that, for \(w=s_\eta p_l\),

Proof. Regard \(a_{uv}=1\) as an edge from \(u\) to \(v\). Irreducibility means that every vertex can reach every other. Some vertex \(v\) has at least two outgoing edges: otherwise every vertex has exactly one successor, and finite strong connectivity makes the matrix a single-cycle permutation matrix. Choose two return paths from \(v\), with different first edges and with no interior visit to \(v\). After each chosen first edge, a shortest path back to \(v\) has the required property. Arbitrary infinite choices between these two return paths give distinct infinite vertex sequences. Visits to \(v\) recover the successive return paths, so there are uncountably many such sequences. Eventually periodic sequences form a countable set, determined by a finite prefix and a finite period. Thus there is a path which is not eventually periodic. Prepending a finite path from \(j\) to \(v\) gives such a path \(x=x_1x_2\cdots\) beginning at \(j\).

Let \(M\) bound both lengths of every listed pair, and let \(D\) bound their nonzero absolute degree differences. For each integer \(1\le d\le D\), non-eventual periodicity supplies an index \(t_d>M\) such that \(x_{t_d}\ne x_{t_d+d}\). Take \(L\) larger than \(M\), the prescribed lower bound and all indices \(t_d+d\). Set \(\eta=x_1\cdots x_L\) and \(l=x_{L+1}\). The allowed transitions and (1.1) give \(w^*w=p_l\) and \(ww^*\le p_j\).

We verify the vanishing algebraically. If either \(\alpha\) or \(\beta\) is not a prefix of \(\eta\), cancellation of the first differing letter makes the compressed word zero. Otherwise write \(\eta=\alpha\mu=\beta\nu\). Both residual words are nonempty because \(L>M\). Cancelling the prefixes gives

An empty original prefix is interpreted as the identity, and gives the same formula. The middle vertex projection either kills one residual word or acts as the identity on it, according to its first letter. If the result is nonzero, cancellation requires the shorter of \(\mu,\nu\) to be a prefix of the longer. Put \(a=|\alpha|\), \(b=|\beta|\), and suppose first that \(a>b\), with \(d=a-b\). Prefix comparability forces \(x_{b+t}=x_{a+t}\) for \(1\le t\le L-a\). In particular it forces \(x_{t_d}=x_{t_d+d}\), since \(t_d>M\ge b\) and \(t_d+d\le L\). This contradicts the choice of \(t_d\). The case \(b>a\) interchanges the residual words and gives the same contradiction. This proves (1.6). \(\square\)

Theorem 1.2 (Cuntz–Krieger simplicity). If \(A\) is irreducible and is not a permutation matrix, \(\mathcal O_A\) is simple.

Proof. Let \(a\ne0\) be positive. Faithfulness of \(E_\gamma\) gives \(\|E_\gamma(a)\|>0\); rescale \(a\) so that this norm is one. Choose a self-adjoint normal-word polynomial \(b\) with \(\|a-b\|<\varepsilon\), where \(0<\varepsilon<1/4\). Its degree-zero part \(b_0=E_\gamma(b)\) belongs to some \(F_r\). The largest eigenvalue of this finite-dimensional self-adjoint element is at least \(1-\varepsilon\): the largest spectral value of the positive element \(E_\gamma(a)\) is one, and norm perturbation changes the largest spectral value by at most the perturbation norm. Choose a block \(j\) and a unit eigenvector \((c_\mu)_\mu\) there, with eigenvalue \(\lambda\ge1-\varepsilon\). The matrix-unit calculation (1.2) shows that

At stage zero use \(x=p_j\). All summands of \(x\) have the same gauge degree \(r\), so \(x^*(b-b_0)x\) is again a finite normal-word polynomial of nonzero degrees. Apply Lemma 1.1 to its finitely many terms. For the resulting \(w\), put \(y=xw\). Equations (1.6) and (1.8) give

The positive element \(y^*ay\) therefore is invertible in the unital corner \(p_l\mathcal O_Ap_l\), whose identity is \(p_l\), because \(\lambda-\varepsilon>0\). Functional calculus in that corner gives \(z=y(y^*ay)^{-1/2}\), and \(z^*az=p_l\). Thus the ideal generated by \(a\) contains a vertex projection.

For each vertex \(i\), irreducibility supplies a path \(l=i_0,i_1,\ldots,i_m=i\). If \(m>0\), the partial isometry \(t=s_{i_0}\cdots s_{i_{m-1}}p_i\) has \(t^*t=p_i\) and \(tt^*\le p_l\), hence \(t^*p_lt=p_i\). For \(i=l\), the projection is already present. The ideal therefore contains every \(p_i\), and their sum is one. Every nonzero closed two-sided ideal contains a nonzero positive element, so the algebra is simple. \(\square\)

This proves the full irreducible, nonpermutation statement credited to Cuntz and Krieger in [Blackadar 1998, Exercise 10.11.9(a)]. It uses the core and faithful gauge average already proved above, together with the explicit finite path compression. No simplicity theorem is imported. The K-theory computation below also applies without irreducibility or simplicity. For example, \(A=(1)\) makes its generator unitary, so universality gives \(C(\mathbb T)\), which is not simple.

2. A stable AF crossed-product model

Set \(D=\mathcal O_A\rtimes_\gamma\mathbb T\), with multiplier unitaries \(\lambda_z\), and let

The projections \(h_k\) belong to \(D\), since \(\mathcal O_A\) is unital. The exact Haar-corner theorem KT-CP-09, Theorem 9.6 identifies \(h_0Dh_0\) with \(F_A\), by \(a\mapsto ah_0\). We must additionally prove that this corner is full.

Covariance gives \(h_k s_i=s_i h_{k-1}\). Therefore

Every column sum is positive, so \(c\) is invertible. Starting with \(h_0\), the ideal it generates contains all positive \(h_k\) by the first equation and all negative ones by the second, multiplying by \(c^{-1}\). The span of \(a h_k\), for \(a\in\mathcal O_A\), is dense in \(D\), by Fourier approximation to continuous crossed-product kernels. Hence the ideal is all of \(D\).

The full-corner Morita map identifies \(K_*(F_A)\) with \(K_*(D)\). Both algebras are separable and thus σ-unital. The precise stable-isomorphism theorem Stable isomorphism and the Brown–Green–Rieffel theorem, Theorem 2.1 now gives

The right side is stable AF: tensor the finite stages (1.4) with increasing finite matrix corners of \(\mathcal K\) and take their dense union.

Choose the positive dual generator \(\sigma\), characterized by \(\sigma(\lambda_z)=z\lambda_z\) and \(\sigma(a)=a\). Then \(\sigma(h_k)=h_{k-1}\). Set \(\beta=\sigma\otimes1\) on \(B\). The inactive tensor-factor identification and Takai duality KT-CP-08, Lemma 8.2, Theorem 8.3 and Proposition 8.4 yield

This proves the required decomposition; the automorphism acts on an actual stable AF algebra.

Proposition 2.1 (the actual shift). Under \(K_0(D)\cong G_A\), the positive dual map is \(\rho^{-1}\), where

Proof. Both maps respect \([v,r]=[Tv,r+1]\); their compositions are the identity by that same relation. No inverse matrix \(T^{-1}\) is used. For a diagonal matrix unit \(w_{\mu,j}w_{\mu,j}^*\) at length \(r\), the partial isometry \(w_{\mu,j}h_{-r}\) has initial projection \(p_jh_{-r}\) and final projection \(w_{\mu,j}w_{\mu,j}^*h_0\). Thus the stage class \([e_j,r]\) corresponds to \([p_jh_{-r}]\). Applying \(\sigma\) replaces \(-r\) by \(-r-1\), exactly the second map in (2.5). As a check, decomposing \(p_i h_{k+1}\) by the partial isometries \(s_i p_jh_k\) gives \([p_i h_{k+1}]=\sum_j a_{ij}[p_jh_k]\). \(\square\)

3. Kernel, cokernel and the unit

Lemma 3.1. For any integer endomorphism \(T\) of \(\mathbb Z^n\), the stage-zero map induces isomorphisms

Proof. In the cokernel on the right, \(\rho\) becomes the identity. Hence \([v,r]=\rho^{-r}[v,0]\) has the same quotient class as \([v,0]\). The inverse map sends \([v,r]\) to \(v\) modulo \((1-T)\mathbb Z^n\). It respects the direct-limit relation because \(Tv\equiv v\) in that quotient and kills \((1-\rho)G_A\). This proves both surjectivity and injectivity.

For kernels, if \(Tv=v\) and \([v,0]=0\), some \(T^k v=0\); but \(T^k v=v\), so \(v=0\). Conversely, if \([v,r]\) is fixed by \(\rho\), equality in the direct limit means \(T^k(1-T)v=0\) for some \(k\). Set \(w=T^k v\). Then \(Tw=w\), and \([v,r]=[w,r+k]=[w,0]\), since a fixed vector can be moved between all stages. This proves surjectivity onto the kernel. \(\square\)

Theorem 3.2. For the matrix and relations of §1,

Proof. Since \(K_1(B)=0\), the PV sequence for (2.4) gives the cokernel and kernel of \(1-\rho^{-1}\) on \(G_A\). The identity \(1-\rho^{-1}=-\rho^{-1}(1-\rho)\) gives the same image and kernel as \(1-\rho\). Apply Lemma 3.1 and stability.

For the unit, follow the actual Takai map rather than choosing an abstract group isomorphism. In KT-CP-08, formula (8.15), \(\lambda_z\) acts by right translation on \(L^2(\mathbb T)\). Its average \(h_0\) therefore becomes \(1_{\mathcal O_A}\otimes e\), where \(e\) is the rank-one projection onto constant functions. A gauge-fixed vertex element \(p_jh_0\) likewise becomes \(p_j\otimes e\); thus each vertex class corresponds to \(e_j\). The inclusion of the full corner sends its unit to \(h_0\). By (1.5), this is the stage-zero vector \(\mathbf1\); a further rank-one stabilization does not change its K-class. \(\square\)

Smith normal form computes the finitely generated cokernel, including its torsion and unit vector. The kernel is free. If \(\det(1-A^t)\ne0\), the kernel is zero and the cokernel has order \(|\det(1-A^t)|\). If the determinant is zero, both have a free part of positive rank, so both groups are infinite. The all-ones matrix is singular when \(n>1\), yet Lemma 3.1 applies without alteration; its algebra is \(\mathcal O_n\).

Zero rows and columns

The classical matrix convention excludes zero rows and columns. If one instead defines an algebra for every zero–one matrix by exactly (1.1), allowing the zero algebra when the relations force \(1=0\), formula (3.2) remains valid. Here is a reduction that also tracks the unit. It is not a statement about graph-algebra relations that omit a range-sum relation at sinks.

If row \(i\) is zero, then \(q_i=0\), so \(s_i=p_i=0\). Deleting that row and column gives an isomorphic algebra. In the cokernel, column \(i\) of \(1-A^t\) is \(e_i\), so that coordinate vanishes. For the kernel, restriction to the remaining coordinates is invertible: a fixed vector on the smaller matrix extends uniquely by \(v_i=\sum_{j\ne i}a_{ji}v_j\). The unit restricts to the smaller all-ones vector.

If column \(i\) is zero, let \(q=\sum_{j\ne i}p_j\) and let \(A'\) be the matrix with vertex \(i\) removed. A normal word can contain \(i\) only as its first letter; compressing by \(q\) removes those terms. Therefore \(q\mathcal O_Aq\) is generated by the remaining partial isometries and is a quotient of \(\mathcal O_{A'}\). This quotient is injective. In \(M_2(\mathcal O_{A'})\) put \(r=\sum_{j\ne i}a_{ij}p_j'\), \(P=\operatorname{diag}(1,r)\), and represent the generators in the corner \(PM_2(\mathcal O_{A'})P\) by

Their initial and final projections verify (1.1), with identity \(P\). Restriction to the \(q\)-corner is the identity representation of \(\mathcal O_{A'}\), proving injectivity. Moreover \(p_i=s_i q_i s_i^*\), with \(q_i\le q\), proves that \(q\) is full. Morita invariance identifies the K-groups, and the unit corresponds to \([1]+[r]\). Algebraically the cokernel eliminates \(e_i\) by \(e_i=\sum_{j\ne i}a_{ij}e_j\), giving precisely the same unit; a kernel vector has \(v_i=0\), since row \(i\) of \(A^t\) is zero. Repeated deletion reaches a matrix without zero rows or columns, or the empty matrix and zero algebra. These K-identifications and unit calculations prove (3.2) in the extended convention as well.

4. Cantor minimal systems and odometers

Let \(X\) be a Cantor space and \(\varphi:X\to X\) a minimal homeomorphism. Use the convention \(\alpha(f)=f\circ\varphi^{-1}\), and let \(u\) implement \(\alpha\) in \(C(X)\rtimes_\alpha\mathbb Z\). The clopen-partition computation of Lesson 18, Proposition 4.2, gives \(K_0(C(X))=C(X,\mathbb Z)\) by rank and \(K_1(C(X))=0\).

Theorem 4.1. Put \(\varphi_*f=f\circ\varphi^{-1}\), \(H=C(X,\mathbb Z)\), and \(B_X=C(X)\rtimes\mathbb Z\). Then

The unit corresponds to the constant function one. The class \([u]\) generates \(K_1\), up to the common sign of the PV boundary convention.

Proof. The PV sequence reduces to the cokernel and kernel of \(1-\varphi_*\) on continuous integer-valued functions. An invariant function is constant on each orbit. A dense orbit and continuity make it constant on \(X\), so the kernel is exactly \(\mathbb Z1\). PV naturality for the equivariant inclusion of scalars identifies the boundary of \([u]\) with \(\pm1\), as in Lesson 18, §4; hence it is a generator. The coefficient inclusion supplies the stated unit. Replacing \(\varphi_*\) by its inverse leaves the image subgroup unchanged. \(\square\)

For an odometer, take positive integers \(N_r\) with \(N_r\mid N_{r+1}\) and \(N_r\to\infty\), and put \(X=\varprojlim\mathbb Z/N_r\mathbb Z\). Addition of one is minimal: an orbit visits every residue at every finite level, hence meets every cylinder set. Every integer-valued continuous function factors through a finite level, since it has finite image and its clopen level sets are unions of finitely many cylinders.

At level \(r\), the cyclic permutation on \(\mathbb Z^{N_r}\) has coinvariant quotient \(\mathbb Z\), through the coordinate sum: adjacent coordinate differences generate exactly the vectors of sum zero. Refinement repeats each coordinate \(m_r=N_{r+1}/N_r\) times, so the connecting map on these quotients is multiplication by \(m_r\). Quotients commute with this algebraic direct limit: a function or a relation belongs to some finite level. Normalizing the stage integer \(a\) to \(a/N_r\) proves

This crossed product is the Bunce–Deddens algebra for the corresponding supernatural number. Here is the actual level model, extending the dyadic construction of Lesson 18, Exercise 18.4. If \(e_0\) is the cylinder projection of residue zero at level \(N=N_r\), put \(E_{ij}=u^i e_0u^{-j}\), for \(0\le i,j<N\). The disjoint residue projections verify the matrix-unit relations and \(\sum E_{ii}=1\). The corner unitary \(v=e_0u^Ne_0\) generates the corner: in a polynomial compression only powers divisible by \(N\) survive, and finite-level coefficient functions compress to scalars. Its spectrum is all of \(\mathbb T\), because the gauge action sends \(v\) to \(z^Nv\), so its nonempty spectrum is invariant under every circle rotation. Functional calculus and the matrix units therefore identify this level algebra faithfully with \(M_N(C(\mathbb T))\). The levels have dense union. Refinement sends a minimal residue projection to \(m_r\) such projections, so its rank K-map is multiplication by \(m_r\). The matrix for \(u\) is the cyclic companion matrix with corner entry \(v\), of determinant \((-1)^{N-1}v\). Its winding is one at every level; since inclusions fix \(u\), their K1 maps are the identity. Thus the positive cone in (4.2) is its nonnegative part and its scale is its intersection with \([0,1]\): at a finite circle-matrix stage, projections have any constant rank from zero to \(N_r\); stability and projection approximation from Lesson 16 pass these rank calculations to the limit. For \(N_r=2^r\), the group is \(\mathbb Z[1/2]\), with unit one. Uniform measure on residues gives the trace value \(a/N_r\), so the ordered calculation agrees with the trace.

Return towers and the algebra obtained by cutting one orbit

The coinvariant calculation (4.1) gives the groups but does not yet exhibit finite or circle matrix blocks inside a general Cantor crossed product. We supply that construction, following the return-tower mechanism of Putnam, as presented in [Weber–Li–Voigt, §12.1–12.2]. It also explains why shrinking a clopen set cannot by itself approximate the implementing unitary in norm.

For a finite clopen partition \(\mathcal P\), write \(C(\mathcal P)\) for the functions constant on its atoms. A tower with roof in a nonempty clopen set \(Y\) has levels \(\varphi^jY_k\), \(1\leq j\leq h_k\), where \(Y_k\subseteq Y\) is a first-return set.

Lemma 4.2 (return towers and refinements). Every nonempty clopen \(Y\subseteq X\) and every finite clopen partition \(\mathcal Q\) admit a tower partition \(\mathcal P\) refining \(\mathcal Q\), with

Proof. Minimality gives uniformly bounded visits to \(Y\) in both time directions. For completeness, finitely many translates \(\varphi^kY\), with \(-M\leq k\leq M\), cover \(X\). Apply this cover to \(\varphi^M x\): some \(\varphi^{M-k}x\) belongs to \(Y\), with \(0\leq M-k\leq2M\). Applying it to \(\varphi^{-M}x\) gives the backwards version. Repeating from a later point gives arbitrarily late visits.

Thus the first positive return \(r(y)\) of \(y\in Y\) is finite. Each set \(\{r=h\}\) is clopen in \(Y\), being the intersection of the return condition at \(h\) and the finitely many nonreturn conditions before it. Continuity into the discrete positive integers and compactness give finitely many heights. Split these sets further by the finite itinerary of \(\varphi^j y\) in \(\mathcal Q\), \(1\leq j\leq h\). Call the resulting nonempty clopen sets \(Y_k\).

Their levels are disjoint. If \(\varphi^j y=\varphi^{j'}y'\) with \(j'>j\), then \(y=\varphi^{j'-j}y'\in Y\), with \(0<j'-j<h_{k'}\), contradicting the first-return definition. Equal levels force equal base points and then the same itinerary set. They cover \(X\): for any \(x\), choose its closest strictly earlier visit \(y\in Y\). There are no intermediate visits, so \(x=\varphi^j y\) with \(1\leq j\leq r(y)\). The last levels are exactly \(Y\), and the first levels are exactly \(\varphi(Y)\). Itinerary refinement gives the stated refinement of \(\mathcal Q\). \(\square\)

For this partition put \(v_Y^0=u1_{X\setminus Y}\). Its superscript distinguishes this cut partial isometry from the cyclic unitary below.

Proposition 4.3 (the orbit-breaking AF algebra). Fix \(y\in X\). Then

is a unital AF-algebra.

Proof. In a tower partition, the operators

are matrix units. One can check the products without manipulating unsupported powers: \(w_i=u^{i-1}1_{P_{k,1}}\) has initial projection \(1_{P_{k,1}}\) and final projection \(1_{P_{k,i}}\), and \(e^k_{ij}=w_iw_j^*\). The final projections for different tower-level pairs are orthogonal, so \(w_j^*w_l\) is zero unless the pairs agree. In the agreeing case it is the initial projection. This proves all matrix-unit and adjoint relations. Each diagonal is nonzero, so each matrix summand is faithfully represented. The diagonals sum to one, and

Conversely, ascending powers of \(v_Y^0\) and the level diagonals give all matrix units, and descending ones follow by adjoints. Hence \(C^*(C(\mathcal P),v_Y^0)=\bigoplus_k M_{h_k}(\mathbb C)\).

Choose decreasing clopen neighborhoods \(Y_r\) of \(y\) with intersection \(\{y\}\), and finite clopen partitions \(\mathcal Q_r\) generating the topology. Apply Lemma 4.2 recursively, refining \(\mathcal Q_r\) and the entire preceding tower partition. Thus the tower partitions \(\mathcal P_r\) are genuinely nested. The associated finite algebras \(F_r\) are nested too: their coefficient spaces are nested, and \(u1_{X\setminus Y_r}=(u1_{X\setminus Y_{r+1}})1_{X\setminus Y_r}\). The last coefficient projection belongs to \(C(\mathcal P_{r+1})\).

The closure of their union contains \(C(X)\), since the partitions generate the topology and locally constant functions uniformly approximate continuous ones. If \(f(y)=0\), then \(f1_{X\setminus Y_r}\to f\) uniformly: outside any neighborhood on which \(|f|\) is small, the nested compact sets \(Y_r\) are eventually absent. Thus the same closure contains \(uf\), by multiplying its cut generator by \(f\) and passing to the limit. Conversely every \(F_r\) lies in \(\mathcal A_y\). This proves (4.4) is exactly their inductive limit. \(\square\)

Its \(K_1\) is zero by Lesson 16, Proposition 1.1. Since \([u]\) is nonzero in \(K_1(B_X)\), \(u\notin\mathcal A_y\). More concretely,

for every \(r\). Shrinking supports give useful coefficient approximation when a function vanishes at \(y\), but never this norm approximation of \(u\).

The return unitary and its full circle spectrum

Lemma 4.4 (inducing on a clopen set). Let \(Y\subseteq X\) be nonempty and clopen. Its first-return map \(\psi(y)=\varphi^{r(y)}y\) is a minimal homeomorphism of the Cantor space \(Y\). Put

This is a unitary in \(1_YB_X1_Y\) implementing \(\psi\), and

with its implementing unitary mapped to \(R_Y\). In particular \(R_Y\) has spectrum \(\mathbb T\) in that corner.

Proof. Continuity of the finite-valued return function makes \(\psi\) continuous. The backwards first-return construction is its inverse, also continuous. A forward \(\varphi\)-orbit visits every open subset of \(Y\), and its successive visits to \(Y\) are precisely the forward \(\psi\)-orbit; thus \(\psi\) is minimal. A clopen subset of a Cantor space is again compact, metrizable, totally disconnected and without isolated points.

The summands in (4.8) have orthogonal initial projections partitioning \(Y\); their final projections also partition \(Y\), by the return-homeomorphism assertion. Thus \(R_Y\) is a corner unitary. Its covariance follows on each return set from the covariance of \(u^h\). This yields a unital homomorphism in (4.9). The source crossed product is simple by the proved minimal-system theorem, Lesson 19, Lemma 6.1, so this nonzero homomorphism is injective.

To prove it is onto, compress the dense crossed-product polynomials by \(1_Y\). For a positive integer \(l\), split \(\{x\in Y:\varphi^l x\in Y\}\) according to the number \(j\), \(1\leq j\leq l\), of returns before time \(l\). These are finitely many clopen sets \(Z_{l,j}\): each condition is determined by the finite list of membership tests at times \(1,\ldots,l\). On each one, \(u^l1_{Z_{l,j}}=R_Y^j1_{Z_{l,j}}\). Negative powers follow by taking adjoints, and power zero is in \(C(Y)\). Multiplying by coefficients shows that every compressed polynomial is in the image. Their density proves surjectivity. The circle gauge action on the source multiplies its implementing unitary by every scalar in \(\mathbb T\). Its nonempty spectrum is therefore invariant under every circle rotation and is all of \(\mathbb T\); faithfulness proves the last assertion. \(\square\)

Approximating the crossed product by circle matrix blocks

Lemma 4.5 (localized cyclic rotation). Given a finite clopen partition \(\mathcal Q\) and an integer \(L\geq1\), there is a unital subalgebra \(D\subseteq B_X\) of the form

which contains \(C(\mathcal Q)\) and a unitary \(\widetilde u\) satisfying \(\|\widetilde u-u\|\leq\pi/L\).

Proof. Choose a point \(y\) and a clopen neighborhood \(Y\) so small that its translates at times \(0,\ldots,L+1\) are pairwise disjoint, and each is contained in one atom of \(\mathcal Q\). This is possible because a minimal homeomorphism of an infinite space has no periodic point. Use Lemma 4.2 to choose a tower partition \(\mathcal P\) refining \(\mathcal Q\). Every height exceeds \(L+1\). In its finite matrix algebra let

This cyclic unitary agrees with \(u\) off \(Y\), and maps \(Y\) to \(\varphi(Y)\).

We keep a roof marker in the smaller finite algebra \(F^0=C^*(C(\mathcal Q),1_Y,v)\). Here is its exact matrix-block description. Label each fine tower by its height and the \(\mathcal Q\)-word of its levels. Towers with the same label are grouped; for each such group, sum their matrix units in (4.5). Those sums are a faithful \(M_h(\mathbb C)\)-system. They generate \(F^0\). To verify the last assertion in both directions, the level sums express the coarse coefficient projections, \(1_Y\), and \(v\). Conversely the commuting diagonal projections \(v^j1_Yv^{-j}\) determine the height of each cyclic orbit by the first return to the marked roof, and \(v^j1_Qv^{-j}\), \(Q\in\mathcal Q\), determine its entire finite level word. Finite intersections and complements of these projections give each grouped roof projection; conjugating it by powers of \(v\) and joining its levels gives the summed matrix units. This also shows that these roof projections are minimal diagonals in \(F^0\), although they can aggregate several finer towers.

Let \(p\) be the grouped roof projection containing \(y\). Choose a nonempty clopen \(Y'\subseteq Y\cap\{p=1\}\) containing \(y\), and a return partition \(\mathcal P'\) with roof \(Y'\) refining \(\mathcal P\). Its finite algebra \(F'\) contains the preceding one by Proposition 4.3's nesting calculation. Let \(w\in F'\) be its cyclic unitary. It agrees with \(u\) off \(Y'\).

Set \(p_j=1_{\varphi^jY}\). The unitary \(d=vw^*\) equals one off \(p_1\): both \(v\) and \(w\) agree with \(u\) off \(Y\) and take \(Y\) onto \(\varphi(Y)\). In the finite-dimensional algebra \(F'\), choose an \(L\)-th root \(z\) of \(d\), choosing root one on its identity eigenspace and eigenvalue \(e^{it/L}\) for \(-\pi<t\leq\pi\). Then

No continuous root on the whole circle is asserted; this is functional calculus on a finite spectrum. With \(Z=X\setminus\bigcup_{j=1}^L\varphi^jY\), put

These summands are unitaries on disjoint corners, so \(W\) is unitary. It belongs to \(F'\): the projections \(p_j\) are sums of the corresponding early levels of \(\mathcal P\), and the conjugations of the \(p_1\)-supported part can use \(v^{j-1}\) instead of \(u^{j-1}\), since none of these early levels reaches a roof. Each \(p_j\) lies in one atom of \(\mathcal Q\), so \(W\) commutes with \(C(\mathcal Q)\). It is identity on \(Y\).

We check the norm estimate on its actual source corners. On \(p_j\), \(1\leq j\leq L\), both cyclic unitaries advance by \(u\). Subtracting \(vW\) and \(Ww\) gives a transported copy of \(z^{L-j}(z-1)\), from source \(p_j\) to range \(p_{j+1}\). On \(1_Y\), \(Ww=z^Lw=dw=v\), while \(vW=v\). On \(1_{Z\setminus Y}\), both products are \(u\): its image under \(\varphi\) avoids all \(p_1,\ldots,p_L\). Thus the error is a sum with orthogonal source corners and orthogonal range corners. Its norm is the maximum of their norms, not their sum. Consequently \(\|WwW^*-v\|\leq\|z-1\|\leq\pi/L\).

The finite algebra \(F^1=W^*F^0W\) still contains \(C(\mathcal Q)\), and its minimal diagonal \(p\) is unchanged because it is supported on \(Y\). The unitary \(t=w^*u\) is identity off \(Y'\), and on that corner is exactly \(R_{Y'}\) of Lemma 4.4: the inverse cyclic wrap after \(u\) takes a roof point through its first positive return. Equivalently, multiplying the terms of (4.11) for the refined towers gives \(t1_{Y'}=\sum_h u^h1_{\{r_{Y'}=h\}}\). Since \(1_{Y'}\leq p\), \(t-1\) is supported inside the minimal diagonal \(p\).

The corner unitary \(ptp\) has spectrum \(\mathbb T\): its \(1_{Y'}\)-corner is \(R_{Y'}\) and its complementary corner is identity. Adjoining it to the matrix units of \(F^1\) therefore replaces just that matrix block by \(M_a(C(\mathbb T))\). Explicitly, if \(p=e_{11}\) after reindexing, the corner-function matrices are \(\sum_{i,j}e_{i1}f_{ij}(ptp)e_{1j}\). The faithful corner calculus and matrix units prove injectivity as well as surjectivity; the other central blocks remain finite matrices. Hence \(D=C^*(F^1,t)\) has (4.10). Finally \(\widetilde u=(W^*vW)t\) is a unitary in \(D\), and \(\|\widetilde u-u\|=\|W^*vW-w\|\leq\pi/L\). This proves every assertion. \(\square\)

Passing from approximation to an actual inductive limit

Lemma 4.6 (repairing finite block generators). Let \(B\) be unital and let \(E\subseteq B\) be a unital finite direct sum of matrix algebras over \(\mathbb C\) or \(C(\mathbb T)\). Given a finite \(\mathcal S\subset E\) and \(\eta>0\), sufficiently good approximation of the matrix units and circle corner unitaries by a unital subalgebra \(C\subset B\) gives a unitary \(V\in B\), as close to one as prescribed, and a unital homomorphism \(\theta:E\to V^*CV\) such that \(\|\theta(x)-x\|<\eta\) for \(x\in\mathcal S\).

Proof. Start with the finite set of diagonal matrix units \(p_t\), summing to one. Approximate them by selfadjoint elements of \(C\). Correct them successively to orthogonal projections in \(C\): compress the next approximation by the complement of the projections already obtained, and take its spectral projection for the part above \(1/2\). At exact data this operation returns \(p_t\); close data have spectrum in disjoint neighborhoods of zero and one, and continuous contour calculus makes the corrected projection tend to \(p_t\). For the last projection take the complement. Its distance from the last \(p_t\) tends to zero too, by the sum identity.

For each matrix block, compress approximations of \(e_{i1}\) by its corrected target and source projections \(q_i,q_1\). Call the resulting operators \(x_i\). For sufficiently close data, \(x_i^*x_i\) is invertible in \(q_1Cq_1\) and \(x_ix_i^*\) is invertible in \(q_iCq_i\), since each tends to the respective corner identity. Therefore \(v_i=x_i(x_i^*x_i)^{-1/2}\) has initial \(q_1\) and final \(q_i\); set \(v_1=q_1\). Their products \(f_{ij}=v_iv_j^*\) give exact matrix units, tending to \(e_{ij}\). Units from different blocks are orthogonal. This constructs the whole finite scalar matrix algebra inside \(C\), with its unit one.

There is a small unitary conjugating the old units to these new ones. Set \(S=\sum_{k,i}f^k_{i1}e^k_{1i}\). Then \(Se^k_{ij}=f^k_{ij}S\), and \(S\to1\) as the approximation improves. For close enough data \(S\) is invertible; its polar unitary \(V=S(S^*S)^{-1/2}\) satisfies the same intertwining relation because \(S^*S\) commutes with all old matrix units. Thus \(V^*CV\) contains the old units exactly, and \(V\to1\).

For a circle summand, let \(a\) be its original generating unitary in the \(e_{11}\)-corner. Conjugate an approximation from \(C\) by \(V^*\), compress by \(e_{11}\), and correct it by polar decomposition in that corner. This gives a corner unitary \(a'\) in \(V^*CV\) tending to \(a\). Functional calculus \(f\mapsto f(a')\), together with the now exact matrix units, defines a homomorphism from its matrix-circle algebra. It need not be injective; its existence only requires that \(a'\) is unitary. Perform this on every circle summand and use the exact units for finite summands. The maps have orthogonal blocks and sum to a unital homomorphism. Noncommutative polynomials in these finitely many generators and their adjoints are dense in \(E\). Contractivity of the homomorphisms and generator convergence therefore give the required approximation on any prescribed finite \(\mathcal S\). \(\square\)

Theorem 4.7 (Cantor crossed products are AT). The crossed product \(B_X\) is isomorphic to a sequential inductive limit of finite sums of matrix algebras over \(C(\mathbb T)\).

Proof. Lemma 4.5 first gives local approximation by circle and finite matrix blocks. Indeed, crossed-product polynomials \(\sum_l f_lu^l\) are dense. Approximate their finitely many coefficients by one \(C(\mathcal Q)\), then use \(\widetilde u\) from that lemma. For unitaries, telescoping gives \(\|\widetilde u^l-u^l\|\leq |l|\|\widetilde u-u\|\) for either sign of \(l\). Taking \(L\) large makes every polynomial in a specified finite collection as close to the block algebra as desired.

Here are the full inductive-limit details. Fix a dense sequence \(b_j\) in the unit ball of the separable algebra \(B_X\), and set \(\epsilon_n=2^{-n}\). Inductively choose unital block subalgebras \(D_n\subset B_X\), homomorphisms \(\theta_n:D_n\to D_{n+1}\), and finite sets \(\mathcal S_n\subset D_n\), so that

Require also that \(\mathcal S_n\) contains an approximant \(c_{n,j}\) with \(\|c_{n,j}-b_j\|<1/n\) for \(j\leq n\). At step \(n+1\), use the local approximation for these \(b_j\) and for the matrix and corner-unitary generators of \(D_n\). Lemma 4.6 conjugates the resulting block algebra by a unitary arbitrarily close to one and supplies \(\theta_n\), close on the prescribed \(\mathcal S_n\). Choose the approximation and conjugation so fine that the new block algebra still approximates \(b_1,\ldots,b_{n+1}\) within \(1/(n+1)\). Include these approximants and \(\theta_n(\mathcal S_n)\) in \(\mathcal S_{n+1}\).

For convergence on every element, include a further countable dense family at a finite pace: enumerate the rational *-polynomials in the generators of each \(D_m\), and at stage \(n\) include the forward images of its first \(n\) polynomials for every \(m\leq n\). There are finitely many such requests at each stage. Every selected polynomial is thereafter propagated by (4.14).

Write \(\theta_{N,n}=\theta_{N-1}\cdots\theta_n\). For a selected element propagated from some stage, its consecutive ambient images differ by at most \(\epsilon_N\); their summable tail makes them Cauchy. For arbitrary \(x\in D_n\), approximate \(x\) by one of the selected dense polynomials. Contractivity of every forward homomorphism bounds the propagated difference by the original approximation error. Thus \(\theta_{N,n}(x)\) is Cauchy for every \(x\), not merely the finite control sets.

Its ambient limit defines homomorphisms \(\Theta_n:D_n\to B_X\), with \(\Theta_{n+1}\theta_n=\Theta_n\). Multiplication and adjoints commute with the norm limits, so the universal property gives a homomorphism \(\Theta:\varinjlim D_n\to B_X\). It is isometric: the norm of the class of \(x\in D_n\) in a C*-inductive limit is \(\lim_N\|\theta_{N,n}(x)\|\), also with kernels in the connecting maps. That is exactly \(\|\Theta_n(x)\|\), by the just-proved ambient norm convergence. Such classes are dense in the limit. It is onto as well: for \(n\geq j\), \[ \|\Theta_n(c_{n,j})-b_j\| \leq\sum_{k=n}^\infty\epsilon_k+1/n \longrightarrow0. \] Its range is closed because it is isometric, and these approximants make the range dense. This supplies the well-definedness, norm and density steps, and proves the isomorphism.

Finally remove finite summands if the AT convention permits only circle summands. Replace each \(M_d(\mathbb C)\) in \(D_n\) by \(M_d(C(\mathbb T))\), obtaining \(E_n\), with evaluation \(\pi_n:E_n\to D_n\) and constant inclusion \(s_n:D_n\to E_n\); use identity on existing circle summands. Set the new connecting map to \(s_{n+1}\theta_n\pi_n\). The maps \(\pi_n\) and \(s_n\) intertwine the systems and induce inverse limit homomorphisms: \(\pi_ns_n=1\), and \(s_n\pi_n\) becomes identity after the next connecting map. Thus the limit is also one with circle summands only, as asserted. \(\square\)

This theorem supplies the structural AT bridge behind the Cantor computation. It does not invoke a classification theorem for arbitrary AT-algebras. That stronger theorem requires a separate proof of its precise order, unit and real-rank hypotheses; a survey statement alone would not supply it.

The order in the Cantor coinvariants

Theorem 4.1 identifies the groups and the unit. The positive cone also has a concrete dynamical description. The inclusion of the algebra obtained by cutting one orbit preserves all of this ordered information, although its degree-one group is zero.

Put \(H=C(X,\mathbb Z)\), \(\alpha(f)=f\circ\varphi^{-1}\), and \[ \begin{gathered} G=H/(1-\alpha)H,\\ G^+=\{[f]:f\in H,\ f\geq0\},\\ e=[1_X]. \end{gathered} \] Let \(\mathcal S_\varphi\) be the invariant states of \(C(X)\): the positive linear functionals \(\lambda\) satisfying \(\lambda(1)=1\) and \(\lambda\alpha=\lambda\). This formulation does not require a measure representation theorem. A state of \((G,G^+,e)\) means an additive map to \(\mathbb R\) that is positive and sends \(e\) to one. Write \(T(B_X)\) for the normalized tracial states of \(B_X\).

Theorem 4.8 (ordered Cantor K-theory). Under the hypotheses of Theorem 4.1, its coefficient-induced isomorphism \[ \kappa:G\longrightarrow K_0(B_X) \] sends \(G^+\) onto \(K_0(B_X)^+\), and sends \(e\) to \([1_{B_X}]\). For every \(y\in X\), inclusion gives an isomorphism of ordered groups with order unit \[ i_*:K_0(\mathcal A_y)\longrightarrow K_0(B_X). \] The ordered group \((G,G^+)\) is a countable simple dimension group and is not cyclic. There are affine bijections \[ \mathcal S_\varphi\longleftrightarrow T(B_X)\longleftrightarrow S(G,e), \] and the trace associated with \(\lambda\) has pairing \[ (\lambda E)_*\kappa([f])=\lambda(f). \] In particular, \[ \begin{gathered} {[f]}\in G^+\setminus\{0\}\\ \Longleftrightarrow\quad\lambda(f)>0\\ \text{for every }\lambda\in\mathcal S_\varphi. \end{gathered} \tag{4.15} \]

Proof. We first record the elementary uniform-sum argument that supplies the order assertion. For \(f\in H\), set \[ S_nf(x)=\sum_{j=0}^{n-1}f(\varphi^j x),\qquad S_0f=0. \] Suppose \(\lambda(f)>0\) for every invariant state. Then \(S_nf(x)>0\) for every \(x\) and every sufficiently large \(n\). If this failed, there would be integers \(n_k\to\infty\) and points \(x_k\) with \(S_{n_k}f(x_k)\leq0\). The orbit averages \[ \lambda_k(a)=\frac1{n_k}\sum_{j=0}^{n_k-1}a(\varphi^j x_k) \] have a subsequence converging on all of \(C(X)\). Here are the compactness details: choose a countable norm-dense set, extract subsequences successively for its bounded scalar values, and take a diagonal subsequence. The norm-one bound extends convergence from that set to every continuous function. Pointwise limits preserve linearity, positivity and the value at one. The endpoint estimate \[ \begin{gathered} |\lambda_k(a\circ\varphi)-\lambda_k(a)|\\ \leq 2\|a\|/n_k. \end{gathered} \] shows that the limit \(\lambda\) is invariant. It has \(\lambda(f)\leq0\), a contradiction. The same extraction from orbit averages of any fixed point shows that \(\mathcal S_\varphi\) is nonempty.

Choose \(N\geq1\) for which \(S_Nf>0\) everywhere and define the continuous integer-valued function \[ h(x)=\min_{0\leq j<N}S_jf(x). \] Since \(h\leq0\) and \(S_Nf>0\), \[ f(x)+h(\varphi x)=\min_{1\leq j\leq N}S_jf(x)\geq h(x). \] Consequently \(g=f+h\circ\varphi-h\geq0\). Its coinvariant class equals \([f]\), because \[ h\circ\varphi-h=(1-\alpha)(h\circ\varphi). \tag{4.16} \] This proves that strict positivity against all invariant states produces a nonnegative integer-valued representative. No uniform ergodic theorem is assumed.

Every invariant state is faithful on \(C(X)\). Indeed, if \(a\geq0\) is nonzero, choose a nonempty clopen \(V\) and \(c>0\) with \(a\geq c1_V\). Finitely many translates of \(V\) cover \(X\), by minimality and compactness. Their characteristic functions sum to at least one. Invariance gives \(m\lambda(1_V)\geq1\), where \(m\) is the number of translates, and hence \(\lambda(a)>0\). Lesson 19, Proposition 1.1 and its faithful gauge average therefore give a faithful tracial state \(\lambda E\) on \(B_X\). Its matrix amplification is faithful as well: for a matrix \(b\), \[ \begin{gathered} (\lambda E)_m(b^*b)\\ =\sum_{i,j}(\lambda E)(b_{ij}^*b_{ij}), \end{gathered} \] so a zero value forces every entry to vanish.

An integer-valued nonnegative function represents a coefficient projection: if \(0\leq g\leq M\), use the diagonal entries \(1_{\{g\geq j\}}\), \(1\leq j\leq M\). Thus \(\kappa(G^+)\subseteq K_0(B_X)^+\). Conversely, let \(x=[p]\) be a nonzero positive class, represented by a projection \(p\in M_m(B_X)\). Use Theorem 4.1 to write \(x=\kappa([f])\). The bounded trace pairing of Lesson 13, Theorem 1.1, including its naturality, gives \[ \lambda(f)=(\lambda E)_m(p)>0 \] for every invariant state. The strict inequality holds because \(p\neq0\) and the amplified trace is faithful. The uniform-sum argument supplies a nonnegative representative of \([f]\). The zero class has representative zero. This proves equality of the cones. It also proves (4.15): a nonzero nonnegative representative has strictly positive value under every invariant state, whereas a coboundary has value zero. The unit assertion is already the coefficient assertion of Theorem 4.1.

We next compare the AF algebra \(\mathcal A_y\). Write it as the increasing union closure of the tower algebras \(F_r\) from Proposition 4.3. Let \[ j:C(X)\longrightarrow\mathcal A_y,\qquad i:\mathcal A_y\longrightarrow B_X \] be the inclusions. The rank identification \(K_0(C(X))=H\) makes \(j_*\) a map \(H\to K_0(\mathcal A_y)\). It is onto: the K-group of each \(F_r\) is generated by one diagonal level projection in each tower, all lying in \(C(X)\); every limit class comes from a finite stage by Lesson 5, Theorem 3.3.

If \(f=(1-\alpha)h\), subtract the constant \(h(y)\) from \(h\). The resulting \(h\) vanishes on a neighborhood of \(y\), since it is continuous and integer-valued. At a sufficiently late stage, \(h\) is constant on every tower level and vanishes on its roof set \(Y_r\). Indeed the nested roof neighborhoods are eventually contained in its zero neighborhood, and the refining coefficient spaces uniformly approximate \(h\). An approximation within \(1/2\) by a level-constant function forces the integer-valued \(h\) to be constant on each level. Expand \(h\) as an integer linear combination of characteristic functions \(q\) of level sets outside \(Y_r\). For each such \(q\), the operator \[ v=(u1_{X\setminus Y_r})q\in F_r \] has \(v^*v=q\) and \(vv^*=uqu^*=\alpha(q)\). Hence \(j_*h=j_*\alpha(h)\), so \(j_*f=0\). Conversely, if \(j_*f=0\), then \(\kappa([f])=i_*j_*f=0\), and the injectivity of \(\kappa\) makes \(f\) a coboundary. Thus \[ \ker j_*=(1-\alpha)H. \] The induced map \(\kappa_y:G\to K_0(\mathcal A_y)\) is a group isomorphism, with \(i_*\kappa_y=\kappa\). It preserves the unit and positivity. Its inverse is positive: if \(a\in K_0(\mathcal A_y)^+\), then \(i_*a\) is positive in \(B_X\); the cone assertion just proved supplies \(g\geq0\) with \(\kappa([g])=i_*a\). Injectivity gives \(a=\kappa_y([g])\), a positive coefficient class. This proves the asserted ordered isomorphism.

For completeness this also gives the dimension-group properties without a classification theorem. The tower stages have ordered groups \((\mathbb Z^{k_r},\mathbb Z_+^{k_r})\), with nonnegative integer connecting matrices. Lesson 5, Theorem 3.3 gives their group and positive-cone limit, which is \(G\) through \(\kappa_y\). It is countable. Positive and negative coordinate parts show directedness. If two positive classes sum to zero, carry their nonnegative representatives to a common later stage where the sum is zero; both representatives then vanish. Thus its cone is proper. If \(nx\geq0\) for an integer \(n\geq1\), carry a representative \(a\) of \(x\) and a nonnegative representative of \(nx\) to a stage where their equality holds. There \(na\geq0\) coordinatewise, so \(a\geq0\); this proves unperforation. For \(x_1,x_2\leq z_1,z_2\), carry all four representatives and the four nonnegative difference representatives to one stage where the defining equations hold. The coordinatewise maximum of the two lower representatives lies below both upper representatives. Its limit class interpolates the four inequalities. These finite-stage arguments allow kernels in the connecting maps.

Every nonzero positive class is an order unit. Choose its nonnegative representative \(g\), which is not identically zero by (4.15). On a nonempty clopen set it is at least one. A finite family of translates of that set covers \(X\), giving \(\sum_{j=1}^m\alpha^{n_j}(g)\geq1\). In the coinvariants this says \(m[g]\geq e\). For arbitrary \(f\in H\), the bound \(-M1\leq f\leq M1\) yields \(-Mm[g]\leq[f]\leq Mm[g]\). This proves simplicity of the ordered group.

It is not cyclic. If it were cyclic, write \(e=m a\) for a generator \(a\) and a nonzero integer \(m\). Any invariant state would take values in \(|m|^{-1}\mathbb Z\) on \(H\). A Cantor space has a partition into more than \(|m|\) nonempty clopen sets. Faithfulness gives each part strictly positive state value, at least \(|m|^{-1}\), while their values sum to one. This is impossible.

Finally restriction sends a trace on \(B_X\) to an invariant state. It determines the trace completely. For \(n\neq0\), absence of periodic points gives a finite clopen partition \(P_l\) with \(P_l\cap\varphi^n(P_l)=\varnothing\). To obtain it, choose a clopen neighborhood with this property at each point, take a finite subcover, and refine to a partition. With \(p_l=1_{P_l}\), cyclicity gives \[ \tau(p_l f u^n)=\tau(p_l f u^n p_l)=0, \] because \(p_lu^np_l=p_l\alpha^n(p_l)u^n=0\). Summing over \(l\) makes \(\tau(fu^n)=0\). On dense Laurent polynomials the trace is consequently \(\lambda E\), and boundedness extends the identity. Proposition 1.1 of Lesson 19 gives the converse extension and proves the first affine bijection.

An invariant state defines a group state by \([f]\mapsto\lambda(f)\). Conversely a group state \(s\) assigns to each clopen partition \((P_l)\) the nonnegative weights \(s([1_{P_l}])\), which sum to one. Define on its locally constant functions \[ \begin{gathered} \lambda\left(\sum_l c_l1_{P_l}\right)\\ =\sum_l c_l s([1_{P_l}]). \end{gathered} \] Refining partitions preserves this formula by additivity. It is a positive norm-one functional on the algebra of locally constant functions, which is dense in \(C(X)\). It extends uniquely to a state. The coinvariant relation makes it invariant on clopen indicators and then on all continuous functions. The two constructions are inverse: every integer-valued continuous function is locally constant. Together with the bounded trace pairing these give the second affine bijection and the displayed pairing formula. \(\square\)

This is the ordered K-theory bridge underlying the Cantor invariant. It does not identify crossed products with isomorphic ordered groups, establish real rank zero, or classify arbitrary AT-algebras. Those require additional theorems.

The ordered orbit-breaking comparison is due to Ian F. Putnam, The C*-algebras associated with minimal homeomorphisms of the Cantor set, Pacific Journal of Mathematics 136 (1989), 329-353, freely readable original paper. The proof here is independently written. The uniform-sum and integer-coboundary argument supplies positive representatives directly, replacing the external positive-lifting result invoked in that paper's proof.

5. Projectionless does not mean trivial K-theory

Here “projectionless” for a unital algebra means that its only projections in the algebra itself are zero and one. It does not assert the same for its matrix algebras.

Proposition 5.1. Let \(X\) be an infinite compact metrizable connected space with \(\check H^1(X;\mathbb Z)=0\), and let \(\varphi\) be minimal. Then \(C(X)\rtimes\mathbb Z\) is simple, unital and projectionless.

Proof. Apply Lesson 19, Theorem 6.2: a minimal system has an invariant probability measure, the resulting trace is faithful, and the connected-space determinant theorem gives trace range exactly \(\mathbb Z\) on \(K_0\). For a projection \(p\) in the unital algebra, \(0\le\tau(p)\le1\). The only integer choices are zero and one. Faithfulness applied to \(p\) or \(1-p\) forces \(p=0\) or \(p=1\). Simplicity is the same theorem's minimal-system conclusion. \(\square\)

Proved existence example. Lesson 19, Lemmas 6.3–6.4 and Theorem 6.5 give the complete Fathi–Herman construction for every compact connected smooth manifold with a locally free circle action, including every odd sphere. In particular the free scalar action gives a smooth minimal diffeomorphism of \(S^3\), isotopic to identity. The proof includes the transverse disks, ambient vector-field motion, finite quotient and equivariant isotopy lift, and the complete category argument. Since \(H^1(S^3;\mathbb Z)=0\), Proposition 5.1 applies. In Exercise 23.5 we compute both K-groups as \(\mathbb Z^2\), and explain why nontrivial matrix projections must exist.

Connes's example. Connes [1982, §12, corollary and following example] uses a cocompact quotient \(V=SL_2(\mathbb R)/\Gamma\) with \(H^1(V;\mathbb Z)=0\), and the time-one lower horocycle map. Lemmas 5.2–5.4 below prove the dynamics and the lattice construction in his specified regular-triangle example, whose three angles are \(\pi/4\). We take \(\Gamma\) to be the full inverse image in \(SL_2\) of the even-reflection subgroup of \(PSL_2\); its quotient is the smooth unit tangent manifold of the doubled triangle orbifold. Proposition 5.1 therefore gives a simple unital algebra with no nontrivial ordinary projections. The proof establishes the required integral first cohomology and retains possible homology torsion. Author's survey.

For clarity, the K-calculation for any such closed oriented three-manifold retains possible torsion. Write \(T_V=H^2(V;\mathbb Z)\), which is finite when \(H^1(V;\mathbb Z)=0\). Then \(K^0(V)=\mathbb Z\oplus T_V\) and \(K^1(V)=\mathbb Z\). Here is the low-dimensional argument. Stable complex bundles on a three-dimensional CW complex are determined by rank and first Chern class: after trivializing on vertices, the obstruction on two-cells is in \(\pi_1(U)=\mathbb Z\), its extension condition is the cellular cocycle condition, and changes of trivialization add coboundaries. There is no further obstruction or invariant on three-cells because \(\pi_2(U)=0\). For stable unitary maps, the determinant invariant is in \(H^1\) and therefore vanishes. A determinant logarithm removes it, leaving maps to stable \(SU\). This group has \(\pi_1=\pi_2=0\) and \(\pi_3=\mathbb Z\), so obstruction theory on cells of dimension at most three identifies these maps with \(H^3(V;\mathbb Z)=\mathbb Z\). The homotopy groups follow from \(SU(2)=S^3\) and the fibrations \(SU(k-1)\to SU(k)\to S^{2k-1}\), then the determinant fibration for \(U\). These classifications apply to K-theory by Lesson 12's bundle and stable-unitary descriptions.

The path of left translations by \(\left(\begin{smallmatrix}1&0\\t&1\end{smallmatrix}\right)\) is a homotopy through homeomorphisms from the identity. Lesson 17, Corollary 4.3 proves that a point-norm path of automorphisms gives isomorphic crossed-product K-groups. For this action on \(C(V)\), joint continuity of translation on the compact manifold gives uniform convergence on \(V\) for each continuous function, hence the required point-norm path. Applying that exact programme proof to the path and the identity action gives

The comparison uses the actual mapping-torus isomorphism and Green/Morita/Thom K-maps proved in Lesson 17; it does not infer a splitting from equal coefficient K-maps. For the identity action the universal commuting relations identify the crossed product with \(C(V\times S^1)\), and the specified coefficient-circle splitting in Lesson 12, Theorem 5.1 gives \(K_i(C(V))\oplus K_{1-i}(C(V))\). An integral homology sphere has \(T_V=0\); the weaker condition \(H^1=0\) alone does not justify deleting \(T_V\). In particular, we have not silently split a torsion extension merely from exactness.

The dynamics and the lattice in Connes's example

We supply both assertions used in the example: the time-one horocycle map is minimal, and a compact quotient with the required integral first cohomology exists. Write \(G=PSL_2(\mathbb R)\), acting on the upper half-plane \(\mathbb H\), and use

All matrices in this argument are understood modulo their common sign. Lower horocycles are conjugate to upper ones by \(\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\), so the conclusion applies to the matrix in Connes's example.

Lemma 5.2 (compact horocycle minimality). If \(\Delta\subseteq G\) is discrete and \(G/\Delta\) is compact, every \(N\)-orbit in \(G/\Delta\) is dense.

Proof. Compactness first excludes nonidentity unipotent elements of \(\Delta\). Choose a compact \(K\) with \(G=K\Delta\). Discreteness gives a neighborhood of identity avoiding nonidentity members of \(\Delta\). By compactness of \(K\), a smaller neighborhood avoids every \(k\delta k^{-1}\), \(k\in K\), \(\delta\in\Delta\setminus\{1\}\). Since every \(g\in G\) is \(k\delta_0\), this neighborhood avoids all conjugates of nonidentity lattice elements. A nonidentity unipotent can be conjugated by diagonal matrices to tend to identity, a contradiction. Thus no \(N\)-orbit has a nonzero period.

The action of \(\Delta\) on the boundary circle is minimal. Here are the geometric details. Inverting \(G=K\Delta\) shows that the orbit \(\Delta i\) is a bounded-distance net in \(\mathbb H\). On a geodesic ray to any boundary point \(\xi\), choose points \(\delta_j i\) within that fixed distance of successive far-out ray points. Then \(\delta_j i\to\xi\). A rotation-diagonal-rotation decomposition of the real matrices, obtained from their polar decomposition, has diagonal parameter tending to infinity. Passing to subsequences of the two rotations shows that \(\delta_j\) converges on the boundary, uniformly off one point \(\zeta\), to the constant \(\xi\). This is also immediate by applying \(\operatorname{diag}(r,r^{-1})\) to the real projective coordinate and letting \(r\to\infty\).

If a nonempty closed invariant boundary set has two points, choose one different from \(\zeta\); its images show that it contains every \(\xi\). It cannot have just one point. Otherwise \(\Delta\) is conjugate into \(B\). Commutators in \(B\) are unipotent, so the absence of nonidentity unipotents makes \(\Delta\) abelian. A nonidentity element of this subgroup is hyperbolic; within \(B\) its centralizer is conjugate to the positive diagonal group. A discrete subgroup there is cyclic. Its orbit in \(\mathbb H\) remains a bounded distance from one axis and cannot be a bounded-distance net in the plane. The trivial subgroup cannot be such a net either. This proves boundary minimality.

Consequently every \(B\)-orbit in \(G/\Delta\) is dense. Indeed \(B\backslash G\) is the boundary circle, by \(g\mapsto g^{-1}\infty\). The inverse image of a closed \(B\)-invariant subset of \(G/\Delta\) is a closed left-\(B\)-saturated, right-\(\Delta\)-invariant subset of \(G\). Its image in \(B\backslash G\) is closed: quotient maps by closed subgroups are open, and the complement of a saturated closed set is a saturated open set. Boundary minimality makes that image the entire circle.

Let \(Y\) be a nonempty minimal compact \(N\)-invariant subset of an orbit closure; compactness and the intersection property of chains give one. Put

The first set is closed by compactness. It satisfies \(N\mathcal D_YN=\mathcal D_Y\). Since \(a_r\) normalizes \(N\), \(a_rY\) is another minimal \(N\)-set; if it meets \(Y\), it equals \(Y\). Thus \(A_Y=\mathcal D_Y\cap\{a_r:r>0\}\) is a closed subgroup.

There exist \(g_j\in\mathcal D_Y\setminus N\) with \(g_j\to1\). Otherwise, at any \(x\in Y\), a sufficiently small quotient chart would have \(Y\) contained in the local \(N\)-plaque through \(x\). That plaque gives an open part of \(Nx\) in \(Y\). Its translates make \(Nx\) open in \(Y\); every other orbit, dense in \(Y\) by minimality, must meet it and hence be the same orbit. The local plaque charts then identify the topology of \(Y=Nx\) with \(\mathbb R\) modulo its period subgroup. That subgroup is discrete and is zero by the first paragraph, so \(Y\) would be homeomorphic to the noncompact line. This contradicts compactness. The quotient chart uses a neighborhood on which \(g\mapsto gx\) is injective, available because the stabilizer is discrete.

Choose determinant-one representatives \(g_j=\left(\begin{smallmatrix}b_j&e_j\\c_j&d_j\end{smallmatrix}\right)\to I\). If \(c_j=0\) infinitely often, \(g_j=a_{b_j}u_{e_j/b_j}\). Because \(g_j\notin N\), \(b_j\ne1\). Then \(a_{b_j}\in A_Y\), with \(b_j\to1\). Under \(r\mapsto\log r\), a closed proper subgroup of \(\mathbb R\) is discrete, so \(A_Y\) is the full diagonal group.

Otherwise \(c_j\ne0\) along a subsequence. Given any \(r>0\), set

The displayed identity follows by multiplication and determinant one. Double-\(N\) invariance and closedness give \(a_r\in\mathcal D_Y\), and hence \(a_r\in A_Y\). Again all diagonal elements preserve \(Y\). Thus \(Y\) is \(B\)-invariant, and the density proved above makes \(Y=G/\Delta\). Every orbit closure contains such a \(Y\), proving the assertion. This is the classical Hedlund mechanism; all of its compactness, boundary and matrix steps have been supplied here. \(\square\)

Lemma 5.3 (every nonzero time is minimal). The map \(x\mapsto u_sx\) of \(G/\Delta\) is minimal for every \(s\ne0\).

Proof. We first give the elementary obstruction for a time-one map of a minimal real flow \(h_t\) on a compact metric space \(X\). Let \(Y\) be a minimal compact \(h_1\)-invariant set. The compact union \(\bigcup_{0\leq t\leq1}h_tY\) is real-flow invariant, hence all of \(X\). Any two translates are minimal sets for \(h_1\), so they are disjoint or equal. The stabilizer \(L=\{t:h_tY=Y\}\) is closed and contains \(\mathbb Z\). If \(L=\mathbb R\), then \(Y=X\). Otherwise \(L=(1/k)\mathbb Z\) for some positive integer \(k\), by the classification of closed subgroups of the line. There is a continuous map \(p:X\to\mathbb R/L\), assigning \(h_tY\) to \(t+L\), and \(p(h_tx)=p(x)+t+L\). To verify continuity, for any converging sequence of points choose representatives \(t_j\in[0,1]\) and \(y_j\in Y\); compact subsequences and continuity of the flow give the asserted fiber of the limit. Uniqueness of the coset then proves continuity. The resulting continuous function \(f(x)=e^{2\pi i k p(x)}\) has modulus one and satisfies

For the horocycle flow, such a nonconstant-frequency function cannot exist. The compact quotient has a finite invariant volume. The bracket matrices in (5.7) have trace zero, so \(\det\operatorname{Ad}(e^Z)=e^{\operatorname{tr}(\operatorname{ad}Z)}=1\); the connected group is generated by exponentials. Thus its left-invariant volume is also right-invariant and descends under lattice translation. Normalize it to mass one. With \(h_t=u_t\), the relation \(a_ru_ta_r^{-1}=u_{r^2t}\) makes \(f(a_rx)\), for \(r>0\), eigenfunctions with distinct nonzero frequencies \(kr^2\). Distinct frequencies are orthogonal in \(L^2(X)\): invariance gives an inner product equal to itself times \(e^{2\pi i(\lambda-\mu)t}\) for every \(t\). These are uncountably many nonzero orthogonal vectors. The \(L^2\) space of a compact smooth manifold is separable, by a countable coordinate cover and approximation by step functions, so this is impossible. Thus the time-one map is minimal. Apply the same argument to the reparameterized flow \(h_t=u_{st}\) for any \(s\ne0\). No ergodicity or mixing theorem is imported. \(\square\)

Lemma 5.4 (Connes's compact lattice with \(H^1=0\)). A regular hyperbolic triangle with angles \(\pi/4\) gives a cocompact discrete subgroup \(\Delta\subseteq PSL_2(\mathbb R)\). For its full inverse image \(\Gamma\subseteq SL_2(\mathbb R)\), the compact smooth oriented manifold \(V=SL_2(\mathbb R)/\Gamma\cong G/\Delta\) satisfies \(H^1(V;\mathbb Z)=0\).

Proof. Regular triangles of increasing size in the disk have equal angles varying continuously from \(\pi/3\) to zero, so one has angles \(\pi/4\). Let \(R_0,R_1,R_2\) be reflections in its sides. We justify the reflection tessellation instead of assuming discreteness. Form the abstract group

Glue one closed copy of the triangle for each \(w\in W\), joining side \(i\) of chamber \(w\) to that of \(wr_i\). Each edge has two adjacent chambers. At a vertex the subgroup generated by the corresponding two reflections has exactly eight elements: the defining relations give at most the dihedral group of order eight, and the actual two reflections act as that group, giving the reverse inequality. The link is therefore a circle of eight angles \(\pi/4\). The glued space is a hyperbolic surface, with no cone singularities.

It is connected. It is simply connected because its dual cells are the Cayley edges \(r_i\) and the eight-sided vertex cells expressing \((r_ir_j)^4\); the only additional relations \(r_i^2\) are edge backtracking. A loop of chambers spells an identity word in the presented group, and successively inserting or deleting these relators contracts it across those cells. Any loop in the surface can be moved off its locally finite vertices and made into such a finite chamber loop.

The metric is complete. The compact chamber has finitely many local geometric models, at its interior, edges and vertices; after the eight-chamber vertex stars and two-chamber edge neighborhoods are included, there is a uniform positive radius on which these models are genuine hyperbolic disks. A Cauchy sequence eventually stays in a smaller such disk, where it converges. Sending chamber \(w\) to \(R_wT\) gives a well-defined local isometry to \(\mathbb H\). Geodesic paths lift for their full lengths, since finite-length lifts are Cauchy at any proposed endpoint. The same disk charts give unique continuation and covering neighborhoods. Thus this local isometry is a surjective covering of the simply connected plane, and is an isometry. This proves that reflected chambers tile the plane, locally finitely with disjoint interiors. The reflection group acts properly, and its compact chamber is a fundamental domain.

Take the index-two even-word subgroup \(\Delta\). It consists of orientation-preserving isometries, is discrete, and has two triangles as a compact fundamental domain. The quotient orbifold \(O=\Delta\backslash\mathbb H\) is the doubled triangle: its underlying space is the sphere and its three cone points have order four. The group is generated by \(R_0R_1,R_1R_2,R_2R_0\), each of order four. Its inverse image \(\Gamma\) under the twofold covering \(SL_2\to G\) is discrete. Moreover \(G/\Delta\cong SL_2/\Gamma\) is compact: over a disk at a cone point its unit tangent bundle is \((D\times S^1)/(\mathbb Z/4)\), with the finite group acting on both factors, and at other points it is the ordinary circle bundle. A finite cover by smaller closed disk charts makes the total space compact. Right translation by a discrete subgroup on the group itself is free and proper, even at its elliptic elements, so this quotient is a smooth oriented three-manifold.

We prove the cohomology assertion on the inverted quotient \(M=\Delta\backslash G\), which is diffeomorphic to \(V\). It is the unit tangent bundle of \(O\); the right rotation circle acts locally freely. In the Lie algebra choose

Their brackets are \([J,X]=Y\), \([J,Y]=-X\), and \([X,Y]=-J\). The left-invariant form \(\eta\) dual to \(J\) descends to \(M\), is rotation-invariant, and is one on the vertical rotation vector field. Its derivative is horizontal and rotation-invariant. At \(i\in\mathbb H\), \(X,Y\) project to the positively oriented orthonormal vectors \(\partial_x,\partial_y\); the bracket formula gives \(d\eta(X,Y)=1\). Hence, for the bundle projection \(\pi\),

Let \(\theta\) be any closed real smooth one-form on \(M\). Average it over the rotation circle. The average \(\theta_0\) differs from \(\theta\) by an exact form: Cartan's formula gives \(\frac{d}{dt}R_t^*\theta=d(R_t^*(\theta(J)))\), and integration in \(t\) proves this assertion. Invariance and the same formula show that \(c=\theta_0(J)\) is constant, since \(M\) is connected. The horizontal invariant form \(\theta_0-c\eta\) descends to an orbifold one-form \(\omega\) on \(O\). Equation (5.8) gives \(d\omega=-c\Omega\). Its integral on \(O\) is zero by Stokes: integrate over the two fundamental triangles, cancel paired edges, and delete small disks at the cone vertices. The remaining boundary integrals tend to zero because \(\omega\) has bounded smooth lifts at those finite-rotation charts. Since \(O\) has strictly positive hyperbolic area, \(c=0\).

Now \(\omega\) is closed. Its lift to the simply connected plane is \(dF\), by integration along paths. Invariance makes \(F(\delta z)-F(z)\) a constant defining a homomorphism \(\Delta\to\mathbb R\). The three finite-order generators force this homomorphism to vanish. Thus \(F\) is invariant and descends to a smooth orbifold function, proving \(\omega\), and then \(\theta\), exact.

This proves \(H^1(M;\mathbb R)=0\), and also the required integral conclusion. To spell out the first-degree topological step, a homomorphism from the fundamental group to \(\mathbb R\) is realized by a closed one-form: on the universal cover, choose a smooth nonnegative cutoff whose translates sum to one, and set \(F(z)=\sum_\gamma h(\gamma)\chi(\gamma^{-1}z)\). Properness makes the sum locally finite and \(F(\delta z)=h(\delta)+F(z)\); \(dF\) descends and has period homomorphism \(h\). Exactness of all closed one-forms therefore makes all such real homomorphisms zero. The group \(H^1(M;\mathbb Z)\) is \(\operatorname{Hom}(\pi_1(M),\mathbb Z)\), by the first-degree universal coefficient and Hurewicz identifications, and injects into these real homomorphisms. It too is zero. This asserts vanishing first cohomology; it does not erase possible torsion in first homology. \(\square\)

The prescribed lower time-one matrix is consequently a smooth minimal diffeomorphism on this \(V\). Lemmas 5.2–5.4 and Proposition 5.1 prove Connes's projectionless example with precisely its required cohomology hypothesis. The existing K-calculation (5.1) retains \(H^2(V;\mathbb Z)\), so no integral homology-sphere conclusion has been inserted into it.

Blackadar's AF mapping-torus construction

The preceding examples use minimal dynamics. Blackadar's original construction instead makes projectionlessness a fixed-point condition on an AF dimension group, then makes the algebra simple by repeatedly folding an interval. We give the complete construction and its ordered \(K_0\), following [Blackadar 1981, §§2–4]. This also supplies examples with several traces or prescribed infinitesimals.

We use the full AF realization proofs in Lesson 16, Theorems 5.4–5.5, and the scaled classification and projection cancellation stated there at their exact programme proof locators. The following lemma supplies the automorphism path needed by the construction.

Lemma 5.5 (paths of AF automorphisms). Let \(B\) be a separable unital AF-algebra. If \(\gamma\in\operatorname{Aut}(B)\) acts as identity on \(K_0(B)\), it can be joined to identity by a point-norm continuous path of automorphisms.

Proof. First the unitary group of every unital AF-algebra is path connected. Approximate a unitary by an element of a unital finite-dimensional stage; its polar unitary remains in that stage and can be made arbitrarily close. Finite-dimensional diagonalization gives a path from its unit to that unitary. The quotient of two sufficiently close unitaries has a continuous logarithm, which gives the remaining short path. The same argument applies in every matrix algebra.

Every corner \(qBq\) is AF. Indeed, projection approximation and the close-projection unitary move \(q\) to a projection \(q'\) of a finite stage. After that stage, the finite algebras \(q'F_nq'\) increase densely in \(q'Bq'\). Conjugating back proves the assertion. If a finite-dimensional unital algebra \(E\subset B\) has matrix units \(e^k_{ij}\), its relative commutant is isomorphic to \(\bigoplus_k e^k_{11}Be^k_{11}\): send \((b_k)\) to \(\sum_{k,i}e^k_{i1}b_ke^k_{1i}\). Multiplication and the matrix-unit identities prove this isomorphism. Its unitary group is therefore path connected too.

Choose a nested unital finite-dimensional generating sequence \(F_n\). The restrictions of identity and \(\gamma\) to \(F_n\) are exactly unitarily conjugate in \(B\). To see this, each minimal diagonal and its image have the same \(K_0\)-class; AF projection cancellation gives a partial isometry \(w_k\) from \(e^k_{11}\) to \(\gamma(e^k_{11})\). The sum \(\sum_{k,i}\gamma(e^k_{i1})w_ke^k_{1i}\) is a unitary, by its orthogonal initial and final projections, and conjugates every matrix unit correctly. Denote one such unitary by \(U_n\).

Then \(U_n^*U_{n+1}\) commutes with \(F_n\). Join it to identity in that relative commutant and multiply the path by \(U_n\). The resulting path from \(U_n\) to \(U_{n+1}\) implements \(\gamma\) on \(F_n\) throughout. Start with \(F_0=\mathbb C1\) and \(U_0=1\), and concatenate on successive intervals approaching time one. At time one put \(\gamma\). This is point-norm continuous there: for \(b\) close to some \(F_n\), the late path and \(\gamma\) agree on that stage, and their values on \(b\) differ by at most twice its distance to the stage. All earlier times are inner automorphisms. This proves the stated path without requiring norm convergence of the implementing unitaries. \(\square\)

Let \(\mathbb Z_2=\varprojlim_r\mathbb Z/2^r\mathbb Z\), and let \(\mathbb Q_2=\bigcup_{r\geq0}2^{-r}\mathbb Z_2\), with the topology for which these are open compact subgroups. Only its additive group and multiplication by two are needed. The clopen cosets \(a+2^r\mathbb Z_2\), \(r\in\mathbb Z\), form a countable base. A compactly supported continuous integer-valued function is a finite integer linear combination of their characteristic functions. This follows by a finite clopen refinement of its finitely many level sets.

On \(G=C_c(\mathbb Q_2,\mathbb Z)\) define \(\theta\) by giving a coset \(a+2^r\mathbb Z_2\) weight \(2^{-r}\). Refining a coset splits it into two equal-weight cosets, so this defines an additive functional independently of the chosen finite partition. Write

The order here is determined by the integral, rather than pointwise positivity.

Lemma 5.6 (the AF data). There are a simple unital AF-algebra \(B\), a nonzero projection \(p\neq1\), an automorphism \(\sigma\), and a corner isomorphism \(\alpha:B\to pBp\), whose \(K_0\)-data are

There is also a point-norm continuous path \(\alpha_t:B\to pBp\) of isomorphisms with \(\alpha_0=\alpha\) and \(\alpha_1=\sigma^2\alpha\sigma^{-1}\).

Proof. The group \(G\) is countable and torsion free. Its integral range is \(\mathbb Z[1/2]\), dense in \(\mathbb R\). The strict cone is proper, directed and unperforated. Every nonzero positive element is an order unit: its strictly positive integral dominates the integral of any given element after taking sufficiently large multiples.

It has interpolation. If a lower bound equals an upper bound, that element interpolates all four inequalities. Otherwise each of the four upper-minus-lower differences has strictly positive integral. Choose a dyadic number strictly between the largest lower integral and smallest upper integral, and an element of \(G\) with that integral. It is a strict interpolant. Lesson 16, Theorem 5.5 now gives a simple unital AF realization with unit \(u\).

Translation preserves \(\theta\), while \(\theta(\mathsf a g)=\theta(g)/2\), by the coset weights. Hence \(\mathsf s\) is a scaled order automorphism. The interval \([0,u]\) contains \(v\); AF scale realization gives \(p\in B\) with that class. Both \(v\) and \(u-v\) have positive integral, so \(p\) is nonzero and proper.

Its AF corner has ordered group \(G\), with unit \(v\). Here is the positive-class justification. Since \([p]\) is an order unit, every matrix projection class \([q]\) is at most \(N[p]\) for some \(N\). Represent the positive difference by another matrix projection. AF cancellation then makes \(q\) equivalent to a subprojection of \(N\) copies of \(p\), a projection over \(pBp\). Conversely an equivalence between two \(p\)-supported projections has its partial isometry supported between them, so it already belongs to the matrix corner. Thus inclusion is an isomorphism of projection monoids and of their ordered group completions. The corner is AF by Lemma 5.5.

The scaled isomorphism \(\mathsf a:(G,u)\to(G,v)\) therefore lifts to \(\alpha\), by the AF classification theorem. Similarly lift \(\mathsf s\) to \(\sigma\). Because \(\mathsf s(v)=u-v\), cancellation gives a unitary conjugating \(\sigma(p)\) to \(1-p\): implement the equivalences of those projections and of their complements and add the two partial isometries. Conjugating \(\sigma\) by that unitary changes no \(K_0\)-map, and gives (5.10).

Direct substitution gives \(\mathsf s^2\mathsf a=\mathsf a\mathsf s\). Also \(\sigma^2(p)=p\). Thus \(\gamma=\alpha^{-1}\sigma^2\alpha\sigma^{-1}\) is an automorphism of \(B\) acting trivially on \(K_0\). Apply Lemma 5.5 and compose its path with \(\alpha\).

Finally \(B\) has a unique tracial state \(\tau\), with pairing \(\theta\). For a group state, the inequalities \(mu-ng>0\) whenever \(m/n>\theta(g)\), and the reversed inequalities for smaller rational bounds, force its value on \(g\) to be \(\theta(g)\). AF finite-stage trace weights give the group-state/trace correspondence, as proved in Lesson 16, Theorem 5.7. Consequently \(\tau\sigma=\tau\) and \(\tau\alpha_t=\tfrac12\tau\) for every \(t\): the latter is a trace of norm \(\tau(p)=1/2\), and division by that norm gives the unique state. \(\square\)

Lemma 5.7 (the folding system). Set

Then \(\Psi:\Gamma\to\Gamma\) is a unital injective homomorphism. The limit \(A=\varinjlim(\Gamma,\Psi)\) is simple and unital. For the data of Lemma 5.6, both \(\Gamma\) and \(A\) have only the ordinary projections zero and one.

Proof. The two corner maps in (5.11) have orthogonal units \(p\) and \(1-p\). Thus their sum is multiplicative, preserves adjoints and has unit one. Point-norm continuity of \(\alpha_t\), isometry of those maps and continuity of \(f\) give a continuous section. The identities \(\kappa\sigma=\sigma\alpha_0\) and \(\sigma\kappa=\alpha_1\) give

Hence the section lies in \(\Gamma\). The norm of an orthogonal corner sum is the maximum of its two norms. As \(t\) varies, its two arguments cover both halves of the interval. Thus \(\|\Psi f\|=\|f\|\), proving injectivity.

We prove simplicity without assuming that a folding picture implies it. If a positive section \(f\) is nonzero, it is nonzero on an open subinterval \(J\subset(0,1)\). Evaluation of \(\Psi^k f\) at \(t\) is an orthogonal sum of injective corner images of \(f((t+m)/2^k)\), \(0\leq m<2^k\). This description follows inductively by substituting the two affine arguments in (5.11); composing corner isomorphisms preserves their orthogonal source summands. For \(2^{-k}<|J|\), at least one of these equally spaced arguments lies in \(J\), for every \(t\in[0,1]\). Therefore \((\Psi^k f)(t)\neq0\) at every fibre.

A positive section nonzero at every fibre is full in \(\Gamma\). To prove this, each fibre is the simple algebra \(B\), and evaluation is onto: at an interior point use a scalar bump, and at the gluing point use interpolation from \(b\) to \(\sigma(b)\). In a fibre, a finite sum \(c=\sum_i a_i f(t)b_i\) can be chosen within distance less than \(1/2\) of one, because the ideal generated by \(f(t)\) is all of \(B\). With \(a=(a_1,\ldots,a_m)\), matrix positivity gives \(c^*c\leq\|a\|^2\|f(t)\|\sum_i b_i^*f(t)b_i\). The left side is invertible, so the positive sum on the right is bounded below by a positive multiple of one. Lift those \(b_i\) to sections. By continuity the corresponding positive section retains a positive lower bound on a neighborhood of that fibre. Finitely many neighborhoods cover the circle. A subordinate scalar partition of unity, multiplying the respective positive sums, gives an invertible positive section in the ideal of \(f\). Its inverse is continuous, respects the endpoint condition and belongs to \(\Gamma\). Thus that ideal contains one.

Every nonzero ideal of the limit contains a nonzero positive element from a stage. More precisely, approximate a norm-one positive ideal element by a positive stage element \(b\) within \(\varepsilon<1/4\). In the quotient by the ideal, \(b\) has norm less than \(\varepsilon\), so functional calculus puts the nonzero \((b-\varepsilon)_+\) in that ideal. The stage inclusion is faithful. By the preceding argument its image is full at a later stage, hence full in the limit. The ideal is the whole limit.

For projectionlessness of a stage, a projection section has a constant \(K_0(B)\)-class \(g\), by projection homotopy, and \(\mathsf s(g)=g\) by its endpoints. An \(\mathsf s\)-fixed continuous function on \(\mathbb Q_2\) is constant on each coset of \(\mathbb Z_2\): integer translations are dense in that subgroup, and continuity passes invariance to their closure. Compact support leaves only finitely many such cosets. Each has weight one, so \(\theta(g)\) is an integer. An ordinary projection has \(0\leq g\leq u\). If it is neither zero nor one, AF cancellation makes both \(g\) and \(u-g\) nonzero positive; then \(0<\theta(g)<1\), impossible. If its value at one point is zero or one, norm continuity of projections forces that value throughout the interval. Hence the section itself is zero or one.

Finally, any ordinary limit projection is within distance less than \(1/2\) of an ordinary stage projection, by selfadjoint approximation and the spectral cut at \(1/2\), as proved in Lesson 1. The stage projection is zero or one. A projection at distance less than one from either is respectively zero or one. This proves the limit assertion. \(\square\)

Theorem 5.8 (the first Blackadar example). The algebra \(A\) above is separable, simple, unital and stably finite, with a unique tracial state, and

Every projection in \(M_d(A)\) is unitarily equivalent to \(\operatorname{diag}(1,\ldots,1,0,\ldots,0)\), with a uniquely determined number of ones. In particular \(A\) has no proper nonzero ordinary projection.

Proof. The mapping-torus exact sequence of Lesson 17 and \(K_1(B)=0\) identify evaluation as an injective map

It also identifies the positive cone exactly. If \(g\in F\cap G^+\), realize it by a projection \(q\in M_d(B)\). Its image \(\sigma(q)\) has the same class. AF cancellation for \(q\) and its complement gives a unitary sending \(q\) to \(\sigma(q)\). Join that unitary to identity in \(M_d(B)\), using Lemma 5.5; its conjugation path of \(q\) is a projection in \(M_d(\Gamma)\) evaluating to \(g\). Conversely every positive class evaluates positively. Thus (5.14) is an ordered isomorphism and carries the unit to \(u\).

We need the corresponding cancellation assertion in the stage, rather than infer it just from exactness. If projection sections \(f,g\in M_d(\Gamma)\) have the same class, their starting projections are unitarily equivalent in \(M_d(B)\). Join the implementing unitary \(U_0\) to \(\sigma(U_0)\); this gives a unitary section and first makes \(f(0)=g(0)=q\). Both sections then end at \(\sigma(q)\). By successively applying the close-projection polar unitary along finite subdivisions, choose continuous unitaries \(a(t),b(t)\), starting at one, with \(f(t)=a(t)qa(t)^*\) and \(g(t)=b(t)qb(t)^*\). The endpoint \(b(1)^*a(1)\) commutes with \(q\). Its relative commutant is the sum of two AF corners, so there is a path \(z(t)\) inside that commutant from one to \(b(1)^*a(1)\). Now \(b(t)z(t)a(t)^*\) has both endpoints one and conjugates \(f\) to \(g\). It is a unitary in \(M_d(\Gamma)\). This proves the claimed cancellation and same-size unitary equivalence.

Elements of \(F\) are finite sums \(\sum_c n_c1_{c+\mathbb Z_2}\), with \(c\) in the quotient group \(\mathbb Q_2/\mathbb Z_2\). Evaluating (5.11) at zero, and using the homotopy-constant class of \(f(t)\), gives the connecting map

For the second identity the two half-weight cosets are \(2c+2\mathbb Z_2\) and \(2c-1+2\mathbb Z_2\); together they partition \(2c+\mathbb Z_2\). Every coset \(c+\mathbb Z_2\) has finite power-of-two order. Therefore, after finitely many applications of \(T\), a finite sum becomes \((\sum_c n_c)u\). The sum is preserved by \(T\). It defines an onto limit map to \(\mathbb Z\); a zero sum becomes zero at a late stage, proving injectivity. Positive nonzero elements have positive sum, and every positive integer multiple of \(u\) is positive. K-continuity and positive projection continuity from Lesson 5 give exactly (5.13).

For a projection in \(M_d(A)\), move it unitarily to a stage projection. Its integer class is some \(n\) with \(0\leq n\leq d\), since its trace on the coefficient AF fibre is between zero and \(d\). At a later stage the class of that projection becomes \(nu\), by (5.15). The stage cancellation just proved makes it unitarily equivalent there to the indicated diagonal projection. Uniqueness follows from its \(K_0\)-class.

It remains to prove the trace assertion. Any trace on \(\Gamma\) has the form

for a probability measure on the glued circle. Here is a justification. The restriction to central scalar functions defines \(\mu\). On an arc trivializing the mapping torus, a nonnegative scalar function \(h\) with support in that arc defines a positive tracial functional \(b\mapsto\rho(hb)\) on \(B\). Uniqueness of \(\tau\) makes it \(\rho(h)\tau(b)\); local cutoff sections justify its product and positivity identities. Approximate a section by finitely many such locally constant fibre values, using a scalar partition of unity. The approximation is uniform, so continuity proves (5.16) for every section. Conversely that integral is a tracial state.

Write \(g(t)=\tau(f(t))\), a continuous function on the glued circle. The half-trace normalization in Lemma 5.6 gives

These shifted Riemann sums converge uniformly in \(t\) to \(\int_0^1g(s)\,ds\), by uniform continuity. A trace on the limit restricts at a given stage to its later-stage trace composed with \(\Psi^k\). Formula (5.16) and this uniform convergence force that restriction to be the Lebesgue integral. Those integrals are compatible under \(\Psi\), so they also construct a trace on the limit and prove uniqueness.

On a simple unital algebra every tracial state is faithful: its zero-square kernel is a closed two-sided ideal, and does not contain one. Its matrix traces are faithful too. For an isometry in any \(M_d(A)\), cyclicity gives trace zero on its missing range projection; faithfulness makes that projection zero. Hence \(A\) is stably finite. Separability follows from that of \(B\), its continuous-section algebra and a sequential limit. \(\square\)

Theorem 5.9 (prescribed ordered group and extreme traces). Let \(X\) be a nonempty compact metrizable totally disconnected space, and let \(N\) be a countable torsion-free abelian group. There is a separable simple unital stably finite algebra \(A_{X,N}\), with only the ordinary projections zero and one, whose ordered \(K_0\) and unit are

Here \(P_H=\{(h,n):h(x)>0\text{ for every }x\in X\}\).

Its tracial state space is affinely homeomorphic to the probability measures on \(X\). The trace pairing is \((h,n)\mapsto\int_Xh\,d\mu\); in particular its extreme traces are parameterized by \(X\), and the \(N\)-summand is infinitesimal for every trace.

Proof. Replace the AF group \(G\) by

Here \(P_W=\{w:\rho(w)(x)>0\text{ for every }x\in X\}\).

Use the constant \(u\) in the first coordinate and zero in the second as order unit, and similarly for \(v\). Both summands are countable and torsion free: a compact space has finite image under a map into a discrete space, its clopen sets form a countable family here, and compactly supported functions on \(\mathbb Q_2\) have finite coset descriptions. The image of \(\rho\) is the locally constant dyadic-valued functions, uniformly dense in \(C(X,\mathbb R)\).

The strict ordered group is directed and unperforated, and every nonzero positive element is an order unit, using uniform upper and positive lower bounds of \(\rho(w)\). For interpolation, an equal lower and upper group element is already an interpolant. Otherwise all four difference functions are strictly positive. Refine a finite clopen partition on which the four functions are constant; on each atom choose a dyadic value strictly between the largest lower and smallest upper value. Such a locally constant function is in \(\rho(W)\) and any of its lifts is a strict interpolant. The AF realization and classification used in Lemma 5.6 therefore apply to \(W\).

Let \(\mathsf s,\mathsf a\) act by translation and doubling on both coordinates, fibrewise on \(C(X,G)\). They preserve the strict order, satisfy \(\mathsf s^2\mathsf a=\mathsf a\mathsf s\), and take \(u,v\) to the same elements as before. Hence all the corner, path and embedding constructions (5.10)–(5.12) apply unchanged.

For an \(\mathsf s\)-fixed element, the first coordinate is a finite coset sum with integer coefficients depending locally constantly on \(x\); its \(\rho\)-function is consequently integer valued. Thus no fixed group element can lie strictly between zero and \(u\), since that would require an integer strictly between zero and one at every \(x\). The projection argument of Lemma 5.7 applies. Its simplicity argument uses only simplicity of the AF fibre and injectivity of the two corner maps, so it applies as well.

The full evaluation, positive-cone and cancellation arguments of Theorem 5.8 give \(K_0(\Gamma)=\ker(1-\mathsf s:W\to W)\). Its connecting map is again \(\mathsf a+\mathsf s^{-1}\mathsf a\). On every \(\mathbb Z_2\)-coset characteristic function with a coefficient in either coordinate, it doubles the coset and retains the coefficient. There are finitely many cosets for each element, uniformly over \(X\). Iteration therefore eventually puts all support in the zero coset. The sums of the coefficients are exactly an element of \(C(X,\mathbb Z)\oplus N\). They are preserved by the connecting map; a zero pair is killed at a late stage, and every pair is realized already in the zero coset. This proves the group isomorphism. Positivity before and after this folding is exactly strict positivity of the first coefficient-sum function, or the zero element. Positive classes in (5.18) are realized by fixed positive elements of \(W\), so positive continuity proves the entire cone and unit, not just the abstract group.

We supply the trace argument also when there is more than one trace. A group state on \(W\) annihilates \(\ker\rho\) and extends to a positive norm-one functional on \(C(X)\). Indeed, the inequalities \(mu\pm nw>0\) for \(m/n>\|\rho(w)\|\) give \(|s(w)|\leq\|\rho(w)\|\). Density extends its bounded additive functional to a real-linear one; strict positivity and passage to limits give positivity. The Riesz representation theorem identifies it with a probability measure on \(X\). Conversely every such measure gives a group state. The AF stage trace-weight proof in Lesson 16 identifies these states with the traces of \(B\). Denote the point-evaluation trace by \(\tau_x\). The map \(x\mapsto\tau_x\) is weak-star continuous, first on projection classes and finite stages and then by norm density. The group maps give \(\tau_x\sigma=\tau_x\) and \(\tau_x\alpha_t=\tfrac12\tau_x\).

Every trace on the mapping torus is integration of \(\Phi(f)(t,x)=\tau_x(f(t))\) against a probability measure on \(\mathbb T\times X\). To justify this assertion without an assumed disintegration, work on trivializing arcs. A positive scalar cutoff \(h\) gives a positive trace \(b\mapsto\lambda(hb)\) on \(B\), hence a positive measure on \(X\) of mass \(\lambda(h)\), by the preceding AF correspondence. A fine partition of unity approximates any selfadjoint section by these local fibre constants. If \(\|\Phi(f)\|_\infty\leq M\), the constant values have trace functions bounded by \(M+\varepsilon\) on every cutoff support, so \(|\lambda(f)|\leq M\), after letting the approximation error vanish. Likewise \(\Phi(f)\geq0\) implies \(\lambda(f)\geq0\). Thus \(\lambda\) factors through \(\Phi\), as a positive bounded functional. Its image span is dense in \(C(\mathbb T\times X)\): trace functions of AF projection differences contain the dense locally constant dyadic functions on \(X\), and multiplying by scalar cutoffs on arcs and using a partition of unity gives their products with arbitrary continuous circle functions. The factored functional extends uniquely, and Riesz representation gives the asserted measure. Conversely integration of fibre traces is a tracial state.

Formula (5.17) holds with \(\tau_x\), and converges uniformly in both \(t\) and \(x\), by joint continuity on a compact space. The \(X\)-marginal of compatible stage trace measures is constant. To verify this, for \(b\in B\) the section \((1-t)b+t\sigma(b)\) has trace function \(\tau_x(b)\), independent of \(t\); its iterated folding has the same trace function. Such functions span a dense subspace of \(C(X)\), so compatibility forces equality of the marginals. Uniform Riemann-sum convergence now forces each stage restriction to be

These functionals are compatible and construct a limit trace for every \(\mu\). The same dense functions independent of \(t\) prove injectivity of the parameterization. The map is continuous and affine, hence an affine homeomorphism from the compact probability measures to its Hausdorff trace space. Point masses are extreme; any measure not a point mass can be split using a clopen set of mass strictly between zero and one, so these are exactly the extreme points. Simplicity makes each trace faithful, and its matrix traces prove stable finiteness as in Theorem 5.8. Its value on a projection class is the first coefficient sum integrated against \(\mu\), proving the asserted pairing. \(\square\)

For \(X\) a point and \(N=0\), this is Theorem 5.8. For a two-point \(X\) and \(N=0\), it gives \(K_0=\mathbb Z^2\) with strict coordinate order and unit \((1,1)\). This ordered group lacks interpolation: the lower bounds \((0,0),(1,0)\) and upper bounds \((2,1),(3,1)\) satisfy all four inequalities, but no integer pair interpolates them. An interpolant distinct from all four bounds would need its second coordinate strictly between zero and one; none of the bounds itself meets the other same-side inequality. Thus ordered \(K_0\) of a simple stably finite algebra need not be a dimension group. The infinitesimal case in (5.18) likewise shows that an integer trace range alone does not determine \(K_0\).

Proposition 5.10 (nuclearity of the construction). All the algebras constructed in Theorems 5.8–5.9 are nuclear.

Proof. We first give finite-dimensional completely positive approximations for \(\Gamma\). A unital finite-dimensional subalgebra \(F\subset B\) admits a unital completely positive retraction \(E:B\to F\). In each block choose a state \(\varphi_k\) of the \(e^k_{11}\)-corner and set its matrix entries to \(\varphi_k(e^k_{1i}be^k_{j1})\). Compression followed by a state is completely positive: positivity of every matrix amplification follows by evaluating its quadratic forms and then the positive functional. The matrix-unit identities give \(E(1)=1\) and \(E|_F=1_F\). Such states exist, for example from the AF corner trace construction. Thus if finitely many \(b\)'s are near \(F\), then \(\|E(b)-b\|\leq2\operatorname{dist}(b,F)\).

Cover the circle by finitely many arcs trivializing the bundle of AF fibres, and choose a scalar partition of unity \(h_i\) subordinate to sufficiently small arcs. For a prescribed finite set of sections, their fibre values in each trivialization are uniformly close to their values at one chosen point \(t_i\). Choose a finite AF stage \(F_i\) approximating those chosen values and a retraction \(E_i\) as above. Evaluation followed by \(E_i\) gives a unital completely positive map \(\vartheta:\Gamma\to\bigoplus_i F_i\). Conversely, transport an element of \(F_i\) as a constant local fibre section, multiply by \(h_i\), extend by zero, and sum over \(i\). This gives a unital completely positive map \(\eta:\bigoplus_iF_i\to\Gamma\): every matrix amplification is positive fibrewise, and the units sum to one. The local transport respects the gluing by \(\sigma\). Uniform continuity and the retraction estimate show that \(\eta\vartheta\) approximates identity on the prescribed finite set arbitrarily well.

For completeness, these factorizations give equality of the two C*-tensor norms. Completely positive contractions tensored with identity are contractions for both minimum and maximum tensor norms. The minimum estimate is the spatial Stinespring compression. For the maximum estimate one uses the same dilation with a commuting second representation: on the Stinespring pre-Hilbert space, let the second algebra act on the Hilbert-vector factor. It is bounded because its operators commute with every entry of the positive matrix defining the inner product; the inequality \(c^*c\leq\|c\|^21\) gives the norm bound. It commutes with left multiplication by the first algebra, and the Stinespring isometry intertwines its action. Compressing that commuting representation gives the asserted maximum estimate. For a nonunital second algebra, the same construction uses its unitization and then restricts to the ideal.

Let \(C\) be any C*-algebra and \(z=\sum_j f_j\otimes c_j\). Coefficient convergence implies \((\eta\vartheta\otimes1)(z)\to z\) in the maximum norm, since its error is at most \(\sum_j\|\eta\vartheta(f_j)-f_j\|\|c_j\|\). Finite matrix algebras have equal maximum and minimum norms, by the matrix-unit representation argument of Proposition 6.4 in Lesson 22. The two contraction estimates therefore give

Passing to the limit, and using the reverse canonical norm inequality, proves that \(\Gamma\) is nuclear.

Finally the stage inclusions into \(A\) are faithful. For a tensor with its finitely many coefficients in one stage, its maximum norm in \(A\otimes_{\max}C\) is at most its maximum stage norm, whereas its minimum norm equals its minimum stage norm by faithful spatial inclusion. Nuclearity of that stage sandwiches the two ambient norms between equal numbers. Approximate the coefficients of an arbitrary algebraic tensor at a common stage; the elementary cross-norm bound makes this approximation valid in both norms. Hence both norms agree for every \(C\), also for the general construction of Theorem 5.9. This is nuclearity of the limit. \(\square\)

These results reproduce the mathematical construction and properties by independent proofs. They do not determine the degree-one group of these examples or classify all simple projectionless algebras.

6. Free groups: the coefficient sequence and its consequences

Let \(F_n\) have generators \(g_1,\ldots,g_n\), and let their action on a C*-algebra \(C\) be given by automorphisms \(\alpha_j\). Write \(R=C\rtimes_{\alpha,r}F_n\) for the reduced crossed product. For \(x=(x_1,\ldots,x_n)\in K_i(C)^n\), define

Theorem (Pimsner–Voiculescu). There is an exact cyclic sequence

The maps \(K_i(C)\to K_i(R)\) come from the coefficient inclusion. The complete arbitrary-coefficient proof is Theorem 6.4 below; see also Blackadar 1998, Theorem 10.8.1. Equivalently, it gives short exact sequences with left term \(\operatorname{coker}\Sigma_i\) and right term \(\ker\Sigma_{1-i}\).

For \(C=\mathbb C\) with the trivial action, both \(\Sigma_i\) vanish. Since scalar K-theory is \(K_0=\mathbb Z\), \(K_1=0\), exactness gives

Proposition 6.1. There is no projection in \(C_r^*(F_n)\) other than zero and one.

Proof. The canonical trace is \(\tau(a)=\langle a\delta_e,\delta_e\rangle\). On group polynomials its tracial property follows by taking the coefficient of the identity in either product, and extends by continuity. It is faithful: if \(\tau(a^*a)=0\), then \(a\delta_e=0\). Every left regular operator commutes with right translations, so \(a\delta_g=0\) for all \(g\); these vectors span a dense subspace, hence \(a=0\). Equation (6.3) shows that the trace range on K-theory is \(\mathbb Z\), since \(\tau(1)=1\). The interval and faithfulness argument of Proposition 5.1 proves the assertion. \(\square\)

The coefficient sequence: a concrete proof

We prove (6.2) for every coefficient algebra, including nonunital and nonseparable ones. This is the theorem of Pimsner and Voiculescu [PV82]. The proof below supplies the operator constructions and the reduced-norm checks inside this lesson. It uses the elementary difference-map construction proved in Lesson 22, Theorem 6.3, equations (6.11)–(6.13): two homomorphisms into a multiplier algebra which agree modulo an ideal give a natural map into the ideal's K-theory. That construction is homotopy invariant; an orthogonal common summand cancels, and the difference of \(\psi+\sigma,\psi\), with \(\sigma\) ideal-valued, induces \(\sigma_*\). It is an elementary consequence of the cone extension there, rather than an imported KK exactness theorem.

First suppose \(C\) is unital. Put \(G=F_n\), \(H=F_{n-1}\), with \(H=\{e\}\) when \(n=1\), and \(B=C\rtimes_rH\). A reduced word belongs to \(W\) if it is empty or its last letter is different from \(g_n^{-1}\). Thus the last \(g_n\)-block, when present at the end, has positive exponent; words ending in any other generator or its inverse also belong to \(W\). Left multiplication gives

For example, cancellation by a left letter can alter the last letter only when the whole intervening word is cancelled. This proves both identities, including their assertions at the empty word.

On the standard Hilbert module \(E_D=\ell^2(W)\otimes D\), where \(D\) contains \(C\), let \(\delta_wb\) denote its \(w\)-coordinate. Define

The \(L_j\)'s are unitaries and \(S\) is an isometry. Write \(p_w\) for a coordinate projection and \(p=p_e\). Then \(1-SS^*=p\). The covariance identities are \(L_j\rho_D(a)L_j^*=\rho_D(\alpha_j(a))\) and \(\rho_D(a)S=S\rho_D(\alpha_n^{-1}(a))\). Let \(T_n\) be the concrete C*-algebra generated by these operators for \(D=C\).

Lemma 6.2 (the one-defect extension). There is a natural exact sequence

The quotient sends the coefficient, \(L_j\), and \(S\) to the coefficient and the corresponding group unitaries. Under \(\mathcal K(E_C)\cong\mathcal K(\ell^2(W))\otimes C\), the vacuum corner is \(a\mapsto p\rho_C(a)\).

Proof. For a word \(w\), replace positive \(g_n\)'s by \(S\), negative ones by \(S^*\), and the other letters by \(L_j^{\pm1}\); call the resulting operator \(s_w\). For \(w\in W\), its action on the vacuum is \(s_w\delta_eb=\delta_wb\): build the reduced word from its right end, which never leaves \(W\). Thus

is exactly the matrix entry \(a\) from coordinate \(z\) to coordinate \(w\). These entries span \(\mathcal K(E_C)\). This is an ideal in \(T_n\), since each generator and its adjoint carries a matrix entry to another entry or zero. It is precisely the ideal generated by \(p\).

Choose a faithful regular covariant representation \((a,v_g)\) of \((C,G)\) on a Hilbert space \(H_0\). One can take \(H_0=\ell^2(G)\otimes H_C\), with \(H_C\) a faithful representation of \(C\), coefficient at coordinate \(h\) equal to \(\alpha_h^{-1}(a)\), and \(v_g\) left translation. The untwisted regular representation on \(\ell^2(G)\otimes H_0\) is \(a\mapsto1\otimes a\), \(u_g\mapsto\lambda_g\otimes v_g\). The diagonal unitary \(\delta_wh\mapsto\delta_wv_wh\) identifies its \(W\)-compression with (6.5).

Let \(P\) be the projection onto \(W\) in this untwisted model. For each \(g\), \(gW\) differs from \(W\) only in the finite set of words of length at most \(|g|\): on a longer word left cancellation cannot reach the last letter. Therefore \([P,\lambda_g]\) has finitely many nonzero matrix entries. Compression \(\Phi(x)=PxP|_{PH}\) takes group polynomials into \(T_n\). Indeed compression of a product differs from the product of compressions by finitely many matrix entries, already in (6.7). Norm continuity gives \(\Phi(R)\subset T_n\), where \(R=C\rtimes_rG\); modulo the compact-module ideal it is multiplicative and preserves adjoints.

For completeness this quotient is the reduced algebra, with no larger completion hidden in it. Right translation sending \(\delta_w\) to \(\delta_{wg_n^{-m}}\) commutes with the untwisted covariant operators. Conjugating the zero-extended compressions by these translations gives projections onto \(Wg_n^{-m}\), which converge strongly to one: \(wg_n^m\in W\) for every fixed \(w\) and all sufficiently large \(m\). The conjugated generators converge strongly, together with their adjoints, to their uncompressed covariant generators. The same holds for every polynomial and then every element, using uniform norm bounds. This defines a contractive homomorphism \(q_n:T_n\to R\). Translated finite matrix entries converge strongly to zero, so it kills \(\mathcal K(E_C)\). On the generators \(q_n\Phi\) is the identity; conversely the induced map \(R\to T_n/\mathcal K(E_C)\) reaches every generator and is inverse to the induced \(q_n\). This proves exactness. All matrix entries here have coefficients in \(C\); no assertion that they are finite-rank operators on the possibly very large \(H_0\) is needed. \(\square\)

We need a norm criterion to justify the upcoming homotopies. For a covariant coefficient representation \(\mu\), unitaries \(u_j\) for \(j<n\), and an isometry \(s\) for \(g_n\), put \(P_0=1-ss^*\) and use the same word substitution \(\sigma_w\).

Wandering-defect criterion. Suppose \(P_0\sigma_wP_0=0\) for every \(w\in W\setminus\{e\}\). On the span of \(\sigma_wP_0H\), the operators are a representation of \(T_n\). If their unitary representation on the orthogonal complement is bounded by the reduced crossed-product norm, they represent \(T_n\) on the whole space.

Here are the details. Covariance makes \(P_0\) commute with \(\mu(C)\). Each \(\sigma_wP_0\), \(w\in W\), is an isometry on \(P_0H\). When a negative \(g_n\)-letter is applied, the already built nonempty terminal subword has range perpendicular to \(P_0H\), so \(s^*\) is isometric there. The other letters are isometries or unitaries. The ranges for different \(w\)'s are orthogonal: cancel their common initial letters in \(\sigma_w^*\sigma_z\). A cancellation \(s^*s\) is exact; a cancellation \(ss^*=1-P_0\) has an error containing the compression of a nonempty terminal subword by \(P_0\), which is zero. At the first different letters the remaining compression is zero by the assumed wandering property; a terminal negative \(g_n\) instead gives \(s^*P_0=0\). For \(w=z\) the result is \(P_0\). Thus \(\delta_w\xi\mapsto\sigma_w\xi\) is a unitary from the regular \(W\)-model with coefficient representation \(\mu|_{P_0H}\) onto this span. Left multiplication and (6.4) show it is reducing for every generator and adjoint. Its orthogonal complement has \(P_0=0\), so all the operators there are unitary. Lemma 6.2 supplies its representation through the reduced quotient whenever the stated norm bound holds.

We will check that norm bound by regular absorption. If \((a,v_g)\) is regular on \(H_0\) and \(w_g\) is any unitary representation on another Hilbert space, the covariant representation \(a\mapsto1\otimes a\), \(u_g\mapsto w_g\otimes v_g\) has the reduced norm. In regular coordinates \(h\), conjugate the extra factor by \(w_h^{-1}\); this removes \(w_g\) from translation and leaves the diagonal coefficient unchanged. Restricting the regular \(G\)-representation to \(H\) is a sum of regular \(H\)-representations over right cosets, with faithful coefficient representations. The same argument therefore applies to \(B\).

Lemma 6.3 (removing one free generator). The inclusion \(d:B=C\rtimes_rF_{n-1}\to T_n\), using the coefficient and \(L_j\), induces isomorphisms in both K-degrees.

Proof. On \(E_B\), let \(\pi_0\) be (6.5). There is also a representation \(\pi_1\), zero on the vacuum, with the following actions on the other coordinates:

On every remaining nonvacuum coordinate \(\pi_1(L_j)\) is left multiplication by \(g_j\); its vacuum action is zero. Here \(u_j\) is the actual unitary of \(B\). This permutation bypasses the deleted vertex and is unitary on \((1-p)E_B\). Covariance on its exceptional edge follows from \(u_j^{-1}\alpha_j(a)u_j=a\); at all other coordinates it is (6.5).

On its unit support \(1-p\), its \(S\)-defect is \(p_{g_n}\). The orbit of that coordinate under the words of \(W\) is the set \(Wg_n\) of words ending in positive \(g_n\)'s, with the usual orthogonal coordinate ranges. The complementary coordinates end in some \(g_j^{\pm1}\), \(j<n\), and all the operators there are unitary. To check the reduced norm, represent \(B\) faithfully by \(a\mapsto a\otimes1\), \(u_j\mapsto v_j\otimes\lambda_j\) on \(H_0\otimes\ell^2(H)\). Undo the diagonal untwisting above. Each group operator then has the form \(w_j\otimes v_j\), including the exceptional edge in (6.8); the extra \(u_j^{-1}\) becomes an extra \(\lambda_j^{-1}\). The reducing coordinate subspaces have this same factorization. Regular absorption and the wandering-defect criterion prove that \(\pi_1\) is a bounded representation of \(T_n\).

The differences \(\pi_0-\pi_1\) are compact-module operators on each generator. They are respectively a vacuum coefficient, two original edges through the vacuum minus the replacement edge, and the edge \(e\to g_n\). Thus the difference map gives \(\Delta:K_i(T_n)\to K_i(B)\), using matrix stability for \(\mathcal K(E_B)\).

We prove both inverse identities by explicit paths. For any coefficient algebra \(D\) containing unitaries \(b_j\) implementing \(\alpha_j\), let \(R_j(t)\), \(0\leq t\leq\pi/2\), be identity off coordinates \(e,g_j\), and on those two coordinates set, writing \(c=\cos t\), \(h=\sin t\), \(\zeta=e^{2it}\),

Multiplying this matrix by its adjoint gives identity, since the diagonal terms are \(\cos^2t+\sin^2t\) and the two off-diagonal terms cancel. It commutes with the coefficient \(\rho_D(\alpha_j(a))\), whose two entries are \(\alpha_j(a),a\): this is exactly \(b_ja=\alpha_j(a)b_j\). Hence \(R_j(t)L_j\) is still covariant. At the endpoint its edge \(e\to g_j\) becomes the vacuum operator \(b_j\), and its edge \(g_j^{-1}\to e\) becomes \(g_j^{-1}\to g_j\) weighted by \(b_j^*\).

Take \(D=B,b_j=u_j\). These operators give representations of \(B\) throughout the path. After untwisting in the preceding faithful representation, every generator is again \(w_j(t)\otimes v_j\); freeness supplies the unitary representation \(w(t)\), and absorption supplies the reduced bound. The path starts at \(\pi_0d\) and ends at the orthogonal sum of \(\pi_1d\) and the vacuum identity representation of \(B\). Its difference from \(\pi_1d\) is compact on each generator. Homotopy invariance and cancellation of the common summand therefore give \(\Delta d_*=1\).

For the other inverse tensor these representations from \(B\) to \(T_n\), so \(D=T_n\) and \(b_j=L_j\) are now elements in the coefficient algebra of \(E_{T_n}\). The coefficient representation remains \(\rho_{T_n}\). First leave the \(L_j\)'s fixed and change \(S\) to

where \(S_{\!T}\) is the original isometry as an element of \(T_n\). The first term's range avoids both \(e,g_n\); on the vacuum the two displayed terms have orthogonal coordinate ranges and squared norm \(\cos^2t+\sin^2t=1\). Thus \(S_t\) is an isometry. Covariance holds term by term, including \(aS_{\!T}=S_{\!T}\alpha_n^{-1}(a)\).

We check that (6.10) really gives representations of the concrete \(T_n\). Its defect is supported on coordinates \(e,g_n\). Powers \(S_t^k\) send this support into the coordinates \(g_n^\ell\), \(0\leq\ell\leq k+1\). For a nonempty \(w\in W\), either \(w=g_n^k\), \(k>0\), in which case the defect compression of \(S_t^k\) is zero for every isometry, or write \(w=vg_n^k\), \(k\geq0\), where \(v\) is nonempty and ends in a generator different from \(g_n\). The remaining reduced letters send those coordinates into \(vg_n^\ell\). None is \(e\) or \(g_n\), and none of these steps returns to the vacuum; all the operators there are the ordinary shifts. The defect compression is again zero. This proves the wandering condition, also for negative \(g_n\)-letters in \(v\).

At \(t=\pi/2\) keep \(S_f=S(1-p)+p\otimes S_{\!T}\) fixed and change the other generators to \(R_j(s)L_j\) by (6.9), with \(b_j=L_j\) and \(0\leq s\leq\pi/2\). The defect \(P_f\) of \(S_f\) is

The coordinates \(Wg_n\) are reducing and never meet the changed edges. Their defect orbit remains the ordinary orthogonal coordinate orbit. On the complementary coordinates put \(q=1-S_{\!T}S_{\!T}^*\). Starting at the vacuum, a letter \(g_j^{\pm1}\), \(j<n\), either stays there with coefficient \(\sin s\,L_j^{\pm1}\), or exits along the corresponding \(g_j^{\pm1}\)-branch. Once it exits, a reduced word cannot return: the next opposite letter would be a cancellation, excluded by reducedness. All subsequent letters then act as ordinary shifts. A \(g_n^{\pm1}\)-letter on the vacuum acts by \(S_{\!T}\) or \(S_{\!T}^*\). Consequently the vacuum block of a reduced word \(w\) is \((\sin s)^{m(w)}s_w\), where \(m(w)\) counts its letters from the first \(n-1\) generators. For \(w\in W\setminus\{e\}\), \(q s_wq=0\) in the original model: \(s_w\delta_e=\delta_w\), and \(w\ne e\). The other output coordinates cannot meet the vacuum or \(Wg_n\). This proves the wandering condition for both parts of (6.11).

The reduced norm condition in both stages follows by the same absorption check, not just the relations. Represent \(T_n\) faithfully on \(\ell^2(W)\otimes H_0\), and undo the diagonal untwisting on the outer copy of \(W\). Every generator in either stage has the form \(w_j\otimes v_j\), where the extra operator acts on the two word-coordinate factors. Its defect, and its generated reducing span, act only on those factors. On the unitary complement the \(w_j\)'s are unitaries; by freeness they form a representation of \(G\), which absorption bounds by the reduced norm. The wandering-defect criterion therefore applies at every parameter, including the endpoints.

These paths vary in point-norm continuously: they do so on the finitely many generators, then on polynomials, and uniform contractivity extends this to every element. Their differences from the tensor-extended \(\pi_1\) are compact, since each altered generator has finitely many matrix entries with coefficients in \(T_n\). The starting representation is the tensor-extended \(\pi_0\). At the final endpoint it is the orthogonal sum of the tensor-extended \(\pi_1\) and the vacuum identity representation of \(T_n\). Naturality of the difference map identifies the initial K-map with \(d_*\Delta\), and cancellation at the endpoint identifies it with identity. Thus \(d_*\Delta=1\). This proves the lemma in both degrees. No separability or nuclearity was used. \(\square\)

Theorem 6.4 (the full coefficient free-group sequence). Equations (6.1)–(6.2) hold for every action of \(F_n\) on every C*-algebra \(C\).

Proof. For each \(k\), use the one-defect algebra \(T_{n,k}\) with \(g_k\) as the distinguished isometry. Let \(D_n\) be their fibre product over their common reduced quotient \(R\). It has the diagonal coefficient representation \(c_n:C\to D_n\), and isometries \(V_j\), whose \(k\)-coordinate is \(S_k\) when \(j=k\), and the unitary \(L_j\) otherwise. There is an exact sequence

The displayed generators generate the whole fibre product. They reach the quotient; the defect of \(V_k\) is zero in every other coordinate, and its translates as in (6.7) span precisely the \(k\)-th compact ideal. A closed subalgebra containing the kernel and reaching the quotient equals the whole algebra.

We claim \(c_{n*}\) is an isomorphism. The base \(n=1\) is Lemma 6.3, with \(B=C\). For induction, map \(D_{n-1}\) into \(D_n\) using the coefficient and \(V_1,\ldots,V_{n-1}\). This map is bounded and injective as follows. In each coordinate \(k<n\), restrict the word model to the subgroup \(F_{n-1}\). Its orbit containing \(e\) is the earlier one-defect model \(T_{n-1,k}\). Every other orbit is a full subgroup orbit, hence a regular reduced representation. Indeed a word outside \(F_{n-1}\) retains a \(g_n\)-letter after multiplication by that subgroup, so its terminal membership in the distinguished half-tree cannot change. These extra regular representations are already bounded by the earlier Toeplitz quotient, and the base orbit is faithful. In coordinate \(n\) all the restricted generators are unitary, giving the inclusion \(C\rtimes_rF_{n-1}\to T_n\) of Lemma 6.3. These maps have the same reduced image in \(R\), and therefore define the claimed fibre-product map.

Projection onto coordinate \(n\) gives the following morphism of extensions: its lower quotient is \(C\rtimes_rF_{n-1}\), its upper quotient is \(T_n\), and both kernels are the direct sum of the first \(n-1\) compact ideals. The maps on those kernels are their coordinate-corner inclusions, induced by \(W_{n-1,k}\subset W_{n,k}\). Each induces the matrix-stability K-isomorphism: it carries a vacuum corner to the same vacuum corner, and any nonempty coordinate corner is full in the compact matrix algebra. The quotient map is a K-isomorphism by Lemma 6.3. The two six-term sequences and the five lemma therefore make \(K_*(D_{n-1})\to K_*(D_n)\) an isomorphism. Composition with the coefficient inclusion completes the induction.

It remains to identify the first arrow, rather than just the groups. Put \(P_k=1-V_kV_k^*\). The two orthogonal homomorphisms \(a\mapsto P_kc_n(a)\) and \(a\mapsto V_kc_n(\alpha_k^{-1}(a))V_k^*\) sum to \(c_n(a)\). The second has the same K-map as \(c_n\alpha_k^{-1}\): the matrix \(\left(\begin{smallmatrix}V_k&P_k\\0&V_k^*\end{smallmatrix}\right)\) is unitary, since \(V_k^*V_k=1\), \(V_kV_k^*+P_k=1\) and \(P_kV_k=V_k^*P_k=0\). It conjugates \(\operatorname{diag}(a,0)\) to \(\operatorname{diag}(V_kaV_k^*,0)\); on normalized unitaries the missing corner is filled by \(P_k\). Matrix stability and invariance under unitary conjugation give the asserted equality in both degrees. The compact ideal's vacuum inclusion therefore has, after the coefficient identification, the map \(1-\alpha_{k*}^{-1}\). This argument applies to relative projection classes and normalized unitaries, by the orthogonal block-sum rule. The first map of (6.12)'s six-term sequence is thus the sum of these \(n\) maps.

Finally change the \(k\)-th ideal coordinate by the group automorphism \(x_k\mapsto-\alpha_{k*}x_k\). Since \((1-\alpha_{k*}^{-1})(-\alpha_{k*}x_k)=(1-\alpha_{k*})x_k\), the first arrow becomes exactly \(\Sigma_i\) in (6.1). Change the corresponding boundary coordinates by the inverse automorphism. The middle arrow remains the coefficient inclusion, because \(D_n\to R\) composed with \(c_n\) is that inclusion. Exactness of the six-term extension sequence now proves (6.2), with its asserted natural maps.

For nonunital \(C\), extend the action to the forced unitization \(C^\sim\), fixing its scalar copy. The quotient \(C^\sim\to\mathbb C\) and its scalar section induce contractive maps on the reduced crossed products, by their regular representations. Their kernel is exactly \(C\rtimes_rG\), without appealing to exactness of the group: if a kernel element is approximated by a group polynomial \(x\), subtract the section of its scalar quotient. The result has all coefficients in \(C\), and its error is at most twice the original approximation error. This gives a split exact sequence on the crossed products. The word constructions, fibre products, rotations and connecting maps are natural for these equivariant coefficient maps. Thus the unital six-term sequence splits, as a complex of abelian groups, into the scalar sequence and its kernel complex. The latter is exact and is precisely (6.2) for \(C\), since split exactness identifies each kernel with the relevant K-group of \(C\) or \(C\rtimes_rG\). This also proves naturality in the nonunital case. Arbitrary faithful Hilbert spaces were allowed throughout, and the compact ideals used a countable word index with arbitrary coefficient algebra. Hence nonseparability introduces no additional hypothesis. \(\square\)

For trivial scalar coefficients, the classes in (6.3) can also be identified directly. In (6.12), the lift of the \(j\)-th group unitary is \(V_j\); its only defect is the vacuum projection in coordinate \(j\). The index convention \([1-V_j^*V_j]-[1-V_jV_j^*]\) gives \(-e_j\). The coordinate change just used is \(-1\) on scalar K-theory, so the boundary in (6.2) sends \([u_j]\) to \(e_j\). Exactness makes these \(n\) classes a basis of \(K_1(C_r^*(F_n))\), while coefficient inclusion makes \([1]\) the generator of \(K_0\). The scalar full-to-reduced quotient therefore is a K-isomorphism directly by Theorem 7.1 and these generators. Corollary 6.5 below gives its stronger coefficient KK comparison from the tree cycle and descent.

Word averaging and simplicity

The preceding projection argument uses the K-group computation. Simplicity has a direct norm proof, originating with Powers [P]. The foundational lesson Free-group averaging and the compact ideal, Section 1, proves the two-generator case through a word partition. The proof here adapts and checks that CC0 programme treatment [A], including every extra generator in the partition, to give the construction for all \(F_n\), \(n\geq2\). The same estimate also proves uniqueness of the trace.

Lemma (free-word partition and averaging). Given a finite \(F\subset F_n\setminus\{e\}\) and \(N\geq1\), there are a partition \(F_n=C\sqcup D\) and words \(u_1,\ldots,u_N\) with

Consequently, for a group polynomial \(x=\sum_{f\in F}c_f\lambda_f\),

Proof. Choose two different generators \(a,b\) and an integer \(L>\max_{f\in F}|f|\), and put \(h=b^La^L\). Each conjugate \(hfh^{-1}\) begins with \(b\) and ends with \(b^{-1}\). If \(f=a^k\ne e\), its reduced conjugate is \(b^La^kb^{-L}\). Otherwise \(f\) contains a letter different from \(a^{\pm1}\); the reduction of \(a^Lfa^{-L}\) cancels fewer than \(L\) letters at either end, leaving nonempty initial \(a\) and final \(a^{-1}\) blocks. Hence the outer \(b\) blocks survive. This reasoning is unchanged when \(f\) uses any of the other generators.

Let \(C_0\) consist of nonempty words starting with \(a\) or \(a^{-1}\), and put \(D_0=F_n\setminus C_0\), including the empty word. There is no cancellation between \(hfh^{-1}\), ending in \(b^{-1}\), and a word of \(C_0\). Their product starts with \(b\), and therefore lies outside \(C_0\). The sets \(a^jD_0\), \(1\leq j\leq N\), are disjoint: each word is \(a^j\) itself, or has exactly \(j\) initial positive \(a\)'s followed by a letter different from \(a^{\pm1}\). Take

Left multiplication by \(h\) transfers the first disjointness assertion to the one just checked; \(u_jD=a^jD_0\) proves the second.

Let \(P\) be the orthogonal projection onto \(\ell^2(C)\) and \(Q=I-P\). The partition gives \(PxP=0\). Set

The \(Q_j\)'s have pairwise orthogonal ranges, and \((I-Q_j)x_j(I-Q_j)=0\). Thus

For a vector \(\xi\), orthogonality of output ranges gives

For the other sum, the triangle inequality followed by Cauchy–Schwarz gives

The two estimates and the decomposition of \(x_j\) prove the norm bound. No positivity or self-adjointness of \(x\) is required. \(\square\)

Theorem (Powers simplicity and the unique trace). For every \(n\geq2\), \(C_r^*(F_n)\) is simple and its canonical trace is its unique tracial state.

Proof. For every \(y\in C_r^*(F_n)\), \(\tau(y)I\) belongs to the norm-closed convex hull of its group-unitary conjugates. To see this, first take \(\tau(y)=0\). Approximate \(y\) by a polynomial \(p\), then remove its identity coefficient: \(x=p-\tau(p)I\). Since \(\|\tau\|=1\),

Choose this less than \(\varepsilon/2\), and then choose \(N\) with \(2\|x\|/\sqrt N<\varepsilon/2\). The preceding conjugation average has norm less than \(\varepsilon\) when applied to \(y\), since averaging unitary conjugations is contractive. Restoring \(\tau(y)I\) proves the convex-hull assertion.

If \(J\) is a nonzero closed two-sided ideal, it contains a nonzero positive element \(y\). Faithfulness of the canonical trace, proved in Proposition 6.1, gives \(\tau(y)>0\). All its conjugates, their averages and their norm limits lie in \(J\). Hence \(\tau(y)I\in J\), and \(I\in J\), proving simplicity.

Every tracial state \(\sigma\) takes the value \(\sigma(y)\) on each group-unitary conjugate and therefore on each of their averages. Taking the norm limit to \(\tau(y)I\) gives \(\sigma(y)=\tau(y)\) for every \(y\). This proves uniqueness. The argument for simplicity and uniqueness uses no K-theory sequence. Together with (6.3), Proposition 6.1 verifies the Kadison–Kaplansky projection assertion for free groups; it makes no claim for every torsion-free group. \(\square\)

There are also full group algebras. Theorem 7.1 below computes their scalar K-theory and actual unitary generators. The explicit index calculation after Theorem 6.4 identifies the same generators in the reduced algebra. The coefficient comparison has the following stronger form.

Corollary 6.5 (the coefficient quotient in KK). Let \(1\leq n<\infty\), and let \(F_n\) act on a separable C*-algebra \(A\), possibly nonunital. The canonical quotient \[ q_A:A\rtimes F_n\longrightarrow A\rtimes_rF_n \tag{6.13} \] is a KK-equivalence.

Proof. The Cayley tree has trivial vertex and edge stabilizers. In Descent and the K-theory of crossed products, Lemma 9.1 and Theorem 9.2, the rooted-tree coisometry and its explicit representation homotopy give a weakly regular cycle representing \(1\in KK^{F_n}(\mathbb C,\mathbb C)\). Thus \(F_n\) is K-amenable. Theorem 8.2 of that lesson applies because \(F_n\) is countable discrete and \(A\) is separable. It constructs \(z_A\in KK(A\rtimes_rF_n,A\rtimes F_n)\) by factoring the full descended unit cycle through the reduced left representation, and proves \[ \begin{gathered} {}[q_A]\widehat\otimes_{A\rtimes_rF_n}z_A=1_{A\rtimes F_n},\\ z_A\widehat\otimes_{A\rtimes F_n}[q_A]=1_{A\rtimes_rF_n}. \end{gathered} \tag{6.14} \] Both products are descent of the same equivariant unit, with the specified full and reduced coefficient modules. These identities prove the assertion and induce the K-isomorphisms in both degrees. The construction includes nonunital \(A\); separability is the countability hypothesis of this KK formulation. \(\square\)

The same provider proves free-product permanence in Theorem 9.5: a finite or countable free product of countable discrete K-amenable groups is K-amenable. Its proof induces the factor unit cycles and pairs their kernel lines with the edges of the Bass–Serre tree. Amenable groups are K-amenable by the weak regularity of the trivial representation, so this also applies to a free product of copies of \(\mathbb Z\). The direct Cayley-tree proof already suffices for Corollary 6.5.

The Baum–Connes viewpoint packages such computations in an assembly map from equivariant topological K-homology to reduced group-algebra K-theory. The bouquet of \(n\) circles has ordinary K-homology ranks one in even degree and \(n\) in odd degree, matching (6.3). Proving the assembly isomorphism is an additional theorem, not a consequence of that rank match. This is the subject of The Baum–Connes assembly map in the Kasparov-theory course; no result from that planned lesson is required here.

7. Full free products with scalar retractions

The following proof supplies the advanced exercise without borrowing the free-group reduced sequence.

Theorem 7.1. Suppose unital C*-algebras \(A_1,A_2\) have characters \(r_j:A_j\to\mathbb C\). For their full unital free product \(F=A_1*_{\mathbb C}A_2\),

The maps to \(F\) are those induced by its two inclusions.

Proof. Existence uses the supremum C*-norm on the algebraic unital free product. The inclusions are injective: retract to \(A_1\) by the identity on \(A_1\) and the scalar character on \(A_2\), and conversely. Define the pullback

Let \(r_P:P\to\mathbb C\) be that common scalar, \(r_F:F\to\mathbb C\) the induced character, and \(c_P,c_F\) their scalar sections. Define homomorphisms

The formula for \(k\) respects the common scalar algebra, so universality applies. Matrix stability allows us to regard \(f_*\) as a map into \(K_i(F)\). We will prove that its corrected version

is inverse to \(k_*\). Both summands have the displayed domain and codomain.

Let \(R_t=\left(\begin{smallmatrix}\cos t&-\sin t\\\sin t&\cos t\end{smallmatrix}\right)\), for \(0\le t\le\pi/2\). In \(M_2(P)\), the map \(k^{(2)}f\) is the pair of matrices \(\operatorname{diag}(a,c)\), \(\operatorname{diag}(c,b)\), where \(c=r_P(a,b)\). Keep the first matrix fixed and conjugate the second by \(R_t\). The scalar images of both matrices stay \(c1_2\), so this defines a homotopy into the pullback. At the endpoint the pair becomes \(\operatorname{diag}((a,b),(c,c))\). Notice that the pair of conjugating matrices need not itself belong to the pullback; the scalar-compatibility check is what makes the homotopy valid. Additivity and homotopy invariance give

For the other composition, keep \(a\in A_1\) represented by \(\operatorname{diag}(a,r_1(a))\), and represent \(b\in A_2\) by \(R_t\operatorname{diag}(b,r_2(b))R_t^*\). The two representations agree on scalars for every \(t\), so the universal property gives a homotopy of homomorphisms from \(F\) to \(M_2(F)\). Its initial map is \(\operatorname{diag}(1_F,c_Fr_F)\), meaning the block sum of those two homomorphisms; its final map is \(fk\). Hence

Since \(kc_F=c_P\) and \(r_Pk=r_F\), subtracting the scalar correction in (7.4) from (7.5) and (7.6) proves \(k_*h_*=1\) and \(h_*k_*=1\).

Finally the extension of \(P\) by \(\ker r_1\oplus\ker r_2\), with quotient \(\mathbb C\), splits by \(c_P\). Each \(A_j\) likewise splits over scalars. Split exactness therefore gives, in degree zero, the two kernel groups and one common copy of \(\mathbb Z\); in degree one it gives just the two kernel groups, since \(K_1(\mathbb C)=0\). Under \(k_*\), the inclusions of the two \(A_j\) add their kernel parts and their scalar coordinates. The only relation between the two scalar coordinates is their difference \(([1_{A_1}],-[1_{A_2}])\). This proves (7.1), including the asserted natural maps. \(\square\)

The type correction in (7.4) matters: a map \(F\to P\) cannot be subtracted from a map \(P\to M_2(F)\) to form this inverse. The displayed proof fixes the map types throughout.

8. Five exercises with complete solutions

Exercise 23.1 (basic: golden mean). Compute both K-groups and the unit class for \(A=\left(\begin{smallmatrix}1&1\\1&0\end{smallmatrix}\right)\).

Solution. The matrix \(1-A^t=\left(\begin{smallmatrix}0&-1\\-1&1\end{smallmatrix}\right)\) has determinant \(-1\), hence is invertible over \(\mathbb Z\). Explicitly its image contains \(-e_2\) from the first column and \(-e_1+e_2\) from the second, so it contains both basis vectors. Its kernel and cokernel are zero. Theorem 3.2 gives \(K_0=K_1=0\); the unit vector \((1,1)^t\) therefore has zero class. This does not mean the identity projection is zero. The path representation is nonzero; indeed irreducibility and the non-permutation condition also give simplicity by Theorem 1.2. \(\square\)

Exercise 23.2 (intermediate: recover \(\mathcal O_2\)). Use \(A=\left(\begin{smallmatrix}1&1\\1&1\end{smallmatrix}\right)\) and identify its algebra as well as its K-groups.

Solution. Every initial projection is \(p_1+p_2=1\), so both generators are isometries, with orthogonal ranges summing to one. Conversely the two Cuntz isometries satisfy (1.1). The universal maps in both directions fix generators and are inverses, giving \(\mathcal O_A\cong\mathcal O_2\). Here \(1-A^t=\left(\begin{smallmatrix}0&-1\\-1&0\end{smallmatrix}\right)\) is an integer involution, so both K-groups vanish. The matrix \(A\) itself is singular: its singularity causes no difficulty in the direct-limit or shift proof. \(\square\)

Exercise 23.3 (intermediate: odometer dimension group). For the odometer of type \((N_r)\), compute its ordered \(K_0\), unit, scale and \(K_1\). Explain the connecting maps.

Solution. At finite level a rank function is a vector in \(\mathbb Z^{N_r}\). The shift relation identifies neighboring coordinate vectors, making its coinvariants the coordinate-sum group \(\mathbb Z\). Refinement to \(N_{r+1}=m_rN_r\) repeats every coordinate \(m_r\) times, so sums multiply by \(m_r\). The compatible embedding into \(\mathbb Q\) sends the stage integer \(a\) to \(a/N_r\), yielding \(G=\bigcup_rN_r^{-1}\mathbb Z\). The constant rank-one function has sum \(N_r\), hence unit one. In the actual \(M_{N_r}(C(\mathbb T))\) stages of §4, nonnegative ranks give \(G^+=G\cap[0,\infty)\), and ordinary projections have ranks between zero and \(N_r\), giving scale \(G\cap[0,1]\). Projection approximation ensures that no additional positive classes or scale values arise at the limit. The winding class of \(u^{N_r}\) in the corner maps to winding one under refinement, so its limit is \(\mathbb Z\); equivalently Theorem 4.1 identifies the generator as \([u]\). The finite quotient action is transitive, so an invariant probability measure assigns each cylinder mass \(1/N_r\). These values determine the measure, and its trace is exactly the order embedding of \(G\). \(\square\)

Exercise 23.4 (advanced: full free-group algebras). Prove \(K_0(C^*(F_n))=\mathbb Z\) and \(K_1(C^*(F_n))=\mathbb Z^n\) using scalar retractions, and identify the generators.

Solution. A full group C*-algebra represents arbitrary unitary representations. Thus the universal algebra of \(n\) unrestricted unitaries is the unital free product of \(n\) copies of \(C(\mathbb T)=C^*(\mathbb Z)\): the two universal maps send each circle coordinate to its corresponding group generator and are inverse on generators. Evaluation at one is a character on each factor and induces a character on every partial free product. Theorem 7.1 therefore applies inductively. Each circle contributes \(K_0=\mathbb Z[1]\) and \(K_1=\mathbb Z[z]\), by Lessons 9 and 12. At every step the two K0 unit generators are identified, leaving a single copy of \(\mathbb Z\). The K1 groups take direct sums. Its natural-map assertion shows that the \(n\) canonical group unitaries give the independent basis classes. This proves both groups without the reduced free-group PV theorem. The explicit boundary calculation after Theorem 6.4 shows that the quotient to \(C_r^*(F_n)\) carries these same classes to its basis and unit generator. \(\square\)

Exercise 23.5 (advanced: ordinary and matrix projections on \(S^3\)). For a minimal homeomorphism \(\varphi\) of \(S^3\), compute the crossed-product K-groups. Show why its lack of ordinary projections does not imply that its K0 is generated by the unit.

Solution. Minimality on the infinite sphere forbids a fixed point. For \(x\in S^3\subset\mathbb R^4\), the formula

is defined for every \(t\in[0,1]\): a zero numerator would imply \(t=1/2\) by norms and then \(\varphi(x)=x\). It homotopes \(\varphi\) to the antipodal map. Simultaneous rotations by angle \(\pi t\) in two orthogonal planes homotope the identity of \(\mathbb R^4\) to \(-1\), restricting to a homotopy on the sphere. Thus \(\varphi\) is homotopic to the identity as a continuous map; we do not need a path through homeomorphisms here. Both induced K-maps are identity. Lesson 12 and Bott periodicity give \(K_0(C(S^3))=\mathbb Z[1]\) and \(K_1(C(S^3))=\mathbb Z\). PV gives, in each degree, an extension of \(\mathbb Z\) by \(\mathbb Z\). Such an extension splits: choose a lift of one in the quotient and send each integer to its multiple. Therefore both crossed-product groups are \(\mathbb Z^2\). The coefficient injection makes the unit a primitive element of \(K_0\), and Proposition 5.1 supplies projectionlessness and trace range \(\mathbb Z\).

By definition every K0 class is a difference of matrix projection classes. If every such projection were stably equivalent to a diagonal projection with entries zero or one, every K0 class would be a multiple of the unit, contradicting \(\mathbb Z^2\). Hence some matrix projection has a class outside \(\mathbb Z[1]\). No assertion about cancellation is needed. In particular the trace has a nonzero kernel even though it is faithful on positive algebra elements: faithfulness does not mean injectivity on virtual K-classes. \(\square\)

What this lesson does not prove

The integer PV sequence, K-continuity, sphere K-theory and connected-space trace theorem are the precise earlier-course inputs cited above. The Haar corner and Takai maps are KT-CP-09, Theorem 9.6, and KT-CP-08, Lemma 8.2, Theorem 8.3 and Proposition 8.4. The passage from Morita equivalence to stable isomorphism is the Hilbert-module lesson's Theorem 2.1. These inputs are used with their stated hypotheses; fullness, the stationary shift, kernel/cokernel reduction and unit identification are proved here.

Cuntz–Krieger simplicity is proved in Theorem 1.2, with the full path compression in Lemma 1.1. The minimal-sphere existence construction is the full programme proof in Lesson 19, Theorem 6.5. Connes's specified horocycle dynamics and cocompact lattice, including its integral first cohomology, are proved in Lemmas 5.2–5.4. The automorphism-path K comparison is the proved programme result in Lesson 17, Proposition 4.2 and Corollary 4.3. The reduced free-group coefficient PV theorem is proved in Theorem 6.4, with the coefficient arrow, signs, nonunital case and reduced norm all checked. Corollary 6.5 applies the written K-amenability quotient and tree proofs in KT-KK Lesson 18, Theorems 8.2 and 9.2; the more general free-product permanence is its Theorem 9.5. Free-group simplicity and uniqueness of trace are proved in Section 6, by an argument independent of those inputs. The projection consequences and the full free-product calculation are proved above. No Baum–Connes assembly theorem is proved or required.

References

• [Blackadar 1998] Bruce Blackadar, K-Theory for Operator Algebras, second edition, Cambridge University Press, 1998: Corollary 10.5.2; Theorem 10.8.1 and Corollary 10.8.2; Exercises 10.11.4, 10.11.7, 10.11.9 and 10.11.11; Proposition 20.9.2(a) and Theorem 20.9.3. Author's book.

• [Emerson 2024] Heath Emerson, An Introduction to C*-Algebras and Noncommutative Geometry, Birkhäuser, 2024, Corollary 11.5.15. Publisher.

• [Cuntz–Krieger 1980] Joachim Cuntz and Wolfgang Krieger, A class of C*-algebras and topological Markov chains, Inventiones Mathematicae 56 (1980), 251–268. Full text.

• [Fathi–Herman 1977] Albert Fathi and Michael R. Herman, Existence de difféomorphismes minimaux, Astérisque 49 (1977), 37–59, §3.4, Theorem 1 and §3.9(c). Primary text.

• [Connes 1982] Alain Connes, A survey of foliations and operator algebras, Proceedings of Symposia in Pure Mathematics 38, part I (1982), 521–628, §12, especially the projectionless corollary and following example. Author's text.

• [P] R. T. Powers, Simplicity of the C*-algebra associated with the free group on two generators, Duke Mathematical Journal 42 (1975), 151–156. The word-partition averaging method is the source of the simplicity argument. Publisher DOI.

• [A] Free-group averaging and the compact ideal, in Foundations of von Neumann algebras: remaining topics, Section 1, Lemmas 1.1–1.2, Proposition 1.3 and Theorem 1.4. Written by GPT-6.1 Sol (OpenAI), Ultra, September 2026; CC0. The checked adaptation here extends its two-generator partition to every finite \(n\geq2\), checks all word cancellations and both operator-norm estimates, and retains its simplicity and unique-trace argument. Exact programme source revision.

• [McMullen 2020] C. McMullen, Ergodic theory, geometry and dynamics, notes dated 24 December 2020, §9, Theorems 9.3–9.4 and 9.8. Author’s notes. Historical Hedlund double-coset mechanism; Lemma 5.2 writes out the boundary and compactness arguments and the explicit matrix limit. Lemmas 5.3–5.4 supply the time-map and specified lattice/cohomology arguments.

• [PV82] M. Pimsner and D. Voiculescu, K-groups of reduced crossed products by free groups, Journal of Operator Theory 8 (1982), 131–156, §§1–3. Journal’s paper. Origin of the coefficient sequence and one-defect Toeplitz method. Lemmas 6.2–6.3 and Theorem 6.4 supply the complete programme proof, with explicit word-support, absorption and both inverse homotopies; no source prose is reproduced.

Self-checked by the writing AI.

[Weber–Li–Voigt] M. Weber, X. Li and C. Voigt, C*-Algebras and Dynamics, ISem24 lecture notes, 20 February 2021, §12.1–12.2, Lemmas 12.2–12.6, Proposition 12.7 and Theorem 12.8, pp. 145–151. Freely readable author notes. The tower and cyclic-rotation mechanism is credited to this treatment and Putnam. The present proof makes refinement explicit, retains a roof marker to identify finite blocks, proves the induced corner and spectrum step, and includes the finite-generator repair and complete limit argument. All exposition is independently written; no source expression is adapted. The general AT classification discussed in §12.3 is not invoked.

[Blackadar 1981, AF mapping torus] B. Blackadar, A simple unital projectionless C*-algebra, Journal of Operator Theory 5 (1981), 63–71, §§2–4, especially Theorem3.1, Proposition3.2, Propositions4.1–4.2 and Theorems4.9–4.12. Freely readable original paper. Lemma5.5–Proposition5.10 independently prove the automorphism path, AF data, endpoint-compatible folding, full simplicity, projectionlessness, exact orderedK0, trace parameterization, prescribed infinitesimals and nuclearity. Missing arguments referred to earlier papers in the original treatment are supplied here, with exact programme AF prerequisites. No source expression is copied, transcribed or reconstructed.