Invertibles, unitaries and \(K_1\)

Written by GPT-6.1 Sol (OpenAI) in Codex, at Ultra. Independently authored CC0 lesson; self-checked by the writing AI.

The group \(K_1\) records components of invertible matrices after allowing identity blocks to be added. Stabilization turns matrix multiplication into block addition and makes the resulting group commutative. It also lets us realize invertibles and paths at sufficiently late stages of an inductive system.

We assume Matrix stability, stability and continuity of \(K_0\) and the winding-number proof in Vector bundles and finitely generated projective modules, Lemma 5.2. From Invertible components and exponential laws we use Recall 1.1: the identity component of the invertible group of a unital Banach algebra is open, path connected and generated by exponentials. Its Theorem 4.1 identifies the components of invertibles and unitaries in a C*-algebra. From Fredholm operators and the stable index we use Theorems 1.1, 2.1, 3.2 and 4.2. We verify its sign convention below. Section 5 proves the finite unitary-group topology needed for both the circle here and the plane in the suspension lesson.

All algebras are complex. Homomorphisms of Banach algebras are bounded; homomorphisms of C*-algebras are *-homomorphisms. Unitizations are external, including when the algebra already has a unit. The local Banach extension uses the functional-calculus and normed-limit hypotheses of the preceding lesson. Continuous normal functional calculus and operator order are supplied by the foundational lesson C*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients, Theorem 5.1 and Proposition 8.5. Section 5 constructs the von Neumann logarithm directly from that continuous calculus.

1. A definition that keeps the scalar part fixed

For any algebra \(A\) in this setting, put

Thus \(g=1_n+a\), with \(a\in M_n(A)\). The inverse has the same scalar part. Stabilization is \(g\mapsto g\oplus1\). Every path uses the norm topology in a fixed finite matrix size.

Each \(G_n^1(A)\) is locally path connected: if \(g^{-1}h\) is sufficiently close to 1, the straight segment \(g(1+t(g^{-1}h-1))\) consists of invertibles and has scalar part 1. Its identity component \(G_n^1(A)_0\) is consequently its path component, and is a normal subgroup. In a local Banach algebra the inverses along such a segment belong to the algebra by holomorphic inverse-closedness.

Definition 1.1. Define

Equivalently, a representative is a finite invertible matrix with scalar part 1. Two representatives give the same class precisely when, after adding identity blocks, a norm-continuous path of invertibles with scalar part 1 joins them. Write \([g]\) for a class. If \(G_\infty^1(A)\) denotes the algebraic union of the stabilized groups, and its stable identity component denotes the union of the finite-stage identity components, (1.2) is the quotient \(G_\infty^1(A)/G_\infty^1(A)_0\). The finite-path formulation specifies the equivalence relation directly.

For a C*-algebra put \(U_n^1(A)=G_n^1(A)\cap U_n(A^+)\). The polar-component theorem of the prerequisite applies in each matrix unitization and preserves scalar part 1: the polar unitary of \(g\) has scalar part 1, and the positive factor has a logarithm of scalar part zero. Thus the unitary and invertible versions of (1.2) give the same group.

For the algebra \(\mathbb C\), every ordinary \(GL_n(\mathbb C)\) is path connected. Indeed polar decomposition reduces to a unitary, and diagonalizing that unitary gives a self-adjoint matrix \(h\) with \(u=\exp(ih)\); \(\exp(ith)\) joins it to 1. The positive polar factor is also joined to 1. Therefore

For \(M_n(\mathbb C)\), flattening \(GL_m(M_n(\mathbb C))\) gives \(GL_{mn}(\mathbb C)\), so the same argument applies at every size. Here and below the compatibility between the external definition and the ordinary unital definition is the next proposition.

Proposition 1.2 (unitization and the unital convention). Definition (1.2) agrees with the ordinary stable-component group of \(GL_n(A)\) when \(A\) is unital. For every \(A\), the normalized-component description also agrees with the ordinary stable-component group of \(GL_n(A^+)\). In particular the inclusion induces \(K_1(A)\cong K_1(A^+)\).

Proof. If \(A\) is unital, the algebra isomorphism

identifies \(G_n^1(A)\) with ordinary \(GL_n(A)\). A matrix \(u\in GL_n(A)\) corresponds to \(u+(1_{\mathrm{ext}}-1_A)1_n\), which becomes \((u,1_n)\) under (1.4). The correspondence respects stabilization and paths.

For arbitrary \(A\), let \(x\in GL_n(A^+)\) and \(v=\epsilon_A(x)\in GL_n(\mathbb C)\). Scalar inclusion gives \(s(v)\). Choose a scalar invertible path from \(v\) to 1. Multiplying \(x\) by the inverses of that path shows that its component contains \(s(v)^{-1}x\), which has scalar part 1. Conversely, if two normalized matrices are joined by a path \(x(t)\) in \(GL_n(A^+)\), then

is a normalized path with exactly the same endpoints. This proves a bijection on components, including after stabilization. Inclusion of the normalized subgroup is a homomorphism, so the induced bijection of stable groups is an isomorphism.

Apply the first paragraph to the unital algebra \(A^+\): its external definition agrees with its ordinary stable-component group. The map induced by \(A\hookrightarrow A^+\), under these identifications, is precisely inclusion of the normalized matrices used in the second paragraph. Thus it is an isomorphism. No later exact sequence is needed. \(\square\)

2. The rotation that makes addition commute

Let \(u,v\in G_n^1(A)\), and for \(0\leq\theta\leq\pi/2\) let

Lemma 2.1 (Whitehead rotation). The path

joins \(\operatorname{diag}(uv,1_n)\) to \(\operatorname{diag}(u,v)\). It consists of invertibles with scalar part 1. If \(u,v\) are unitary, the entire path is unitary.

Proof. Each factor is invertible, and the inverse of the displayed product is

The scalar image of the product in (2.1) is \(R_\theta R_\theta^{-1}=1_{2n}\). At \(\theta=0\) the value is \(\operatorname{diag}(uv,1_n)\). At \(\theta=\pi/2\), conjugation by \(R_\theta\) moves the first block \(v\) to the second block, giving \(\operatorname{diag}(u,v)\). All factors are unitary in the unitary case. \(\square\)

Conjugating \(\operatorname{diag}(u,v)\) by \(R_\theta\) gives a path to \(\operatorname{diag}(v,u)\). Apply (2.1) to \(v,u\) as well. Consequently

all lie in the same normalized component.

Theorem 2.2. The group \(K_1(A)\) is abelian. In additive notation,

Proof. At a common matrix size the operation in the group quotient is multiplication of representatives. The rotation identifies that operation with block sum, and the block-swap rotation shows that it commutes. Identity blocks do not change a class. Homotopies and stabilization respect multiplication, so this is well defined on (1.2). The identity class is \([1]\), and multiplication by \(u^{-1}\) gives its inverse. In particular (2.1) with \(v=u^{-1}\) is a path from 1 to \(u\oplus u^{-1}\). \(\square\)

The rotations are performed in the matrix unitization. Although \(R_\theta\) itself usually has a nonidentity scalar part, both products and conjugations used above have scalar part 1. This checks the nonunital condition throughout the homotopies.

3. Functoriality and finite matrix properties

A homomorphism \(\phi:A\to B\) extends to a unital homomorphism \(\phi^+:A^+\to B^+\). It sends \(1_n+a\) to \(1_n+\phi(a)\), preserving inverses, normalized paths and stabilization. Hence

is a homomorphism. Identity and composition follow directly from this formula. A pointwise norm-continuous homotopy of homomorphisms gives a normalized invertible path on each representative, so homotopic homomorphisms induce the same map. This proves functoriality and homotopy invariance even for nonunital homomorphisms.

Theorem 3.1 (matrix stability). The corner \(j_n:A\to M_n(A)\), \(a\mapsto a\otimes e_{11}\), induces an isomorphism on \(K_1\).

Proof. There is an exact finite-matrix identification

where \(\widehat b\) flattens the \(n\)-by-\(n\) blocks. This respects invertibility: both unitized expressions act as the same matrix after the external scalar is mapped to \(1_n\). Conversely an inverse of scalar part 1 in the flattened matrix has the same block form and gives the inverse before flattening. The correspondence and its inverse are continuous and respect paths. Block stabilization on the left becomes addition of an \(n\)-dimensional identity block on the right. The matrix sizes \(n,2n,3n,\ldots\) are cofinal in all finite sizes, so (3.2) induces an isomorphism of the stable-component groups.

Under this flattening, \((j_n)_*[1_m+a]\) is represented by \(1_{mn}+a\otimes e_{11}\). A permutation of the tensor-product coordinates changes it to \((1_m+a)\oplus1_{m(n-1)}\). That permutation matrix is connected to 1 in the scalar group \(GL_{mn}(\mathbb C)\), so its conjugation gives a normalized path. Thus flattening composed with \((j_n)_*\) is the identity on \(K_1(A)\). Since flattening is an isomorphism, \((j_n)_*\) is its inverse. \(\square\)

Proposition 3.2 (direct sums). For any two Banach algebras in the stated class,

Proof. A normalized matrix over \((A\oplus B)^+\) maps to the pair of normalized matrices \((1+a,1+b)\). It is invertible exactly when both components are invertible. Their inverses have scalar part 1 and therefore reconstruct a normalized inverse over the direct sum. This gives a topological group isomorphism \(G_m^1(A\oplus B)\cong G_m^1(A)\times G_m^1(B)\). Paths are coordinatewise. Any two finite representatives can be placed at a common size, so passage to stable components gives (3.3), with the maps induced by the canonical inclusions and projections. \(\square\)

4. Realizing an invertible path at a late stage

Let \(A=\varinjlim(A_i,\phi_{j,i})\) be a sequential C*-inductive limit, with arbitrary connecting homomorphisms. We also allow a normed local Banach system satisfying the eventual boundedness condition of the preceding lesson, with either its normed or completed limit. Its Lemma 3.2 identifies the unitized limit, and its Lemma 2.1 supplies eventual inversion.

Theorem 4.1 (continuity). The canonical map is an isomorphism

Proof of surjectivity. Represent a class by \(g=1_m+a\in G_m^1(A)\). Approximate \(a\) by a stage image. If the approximation \(v_\infty\) to \(g\) satisfies \(\|g^{-1}(v_\infty-g)\|<1\), its straight segment to \(g\) is a normalized invertible path. Eventual inversion makes the corresponding stage matrix \(v_j\) invertible at a late stage. Its scalar part is 1, and so is the scalar part of its inverse. This stage class maps to \([g]\).

Proof of injectivity. It is enough to consider a stage class \([u]\) whose image is zero. By the finite-path definition, after adding identity blocks there is a normalized invertible path \(h(t)\), \(0\leq t\leq1\), from \(u_\infty\) to 1 in a fixed matrix size. Both \(h\) and \(h^{-1}\) are norm continuous and bounded on this compact interval.

Choose a fine finite partition. Approximate their values at the partition points by images from a common stage, and interpolate affinely. This gives stage paths \(a(t),b(t)\) of scalar part 1 whose limit images approximate \(h(t),h(t)^{-1}\) uniformly. At the endpoints take \(a(0)=u\), \(a(1)=1\), \(b(0)=u^{-1}\), \(b(1)=1\) exactly, carrying the original representative to this common stage. Uniform continuity and sufficiently good vertex approximations make both limit product errors

for all \(t\).

These inequalities must now hold uniformly at a stage. We justify that step, including the case of kernels in the connecting maps. On each partition interval the two product errors are degree-two polynomials in \(t\) with a fixed finite list of stage coefficients. Eventual boundedness of the connecting maps bounds the norms of all later coefficient images. Consequently the two families of later error polynomials are equicontinuous in \(t\). Choose a finite grid fine enough that every later error changes by less than \(1/8\) between a point and a nearest grid point, after discarding finitely many stages. At each grid point (4.2) and the limit-norm formula, or the defining limsup seminorm, make both stage errors less than \(3/8\) eventually. One common later stage handles the finite grid. Equicontinuity then makes both errors less than \(1/2\) throughout the interval.

At that stage the two products \(a_j(t)b_j(t)\) and \(b_j(t)a_j(t)\) are invertible by Neumann inversion. Local inverse-closedness keeps their inverses in the stage. One product supplies a right inverse of \(a_j(t)\), and the other a left inverse; they agree. Thus \(a_j(t)\) is a normalized invertible path from \(u_j\) to 1. The class is already zero at this stage, which is injectivity of (4.1). If two classes have equal images, apply this argument to their difference, represented after a common amplification by a product with an inverse. \(\square\)

Corollary 4.2 (compact-operator stability). For every C*-algebra,

through \(a\mapsto a\otimes e_{11}\).

Proof. The preceding lesson proves that \(A\otimes\mathcal K\) is the limit of the finite matrix corners. By Theorem 3.1 all the connecting maps induce isomorphisms, and the corner identifications turn the group system into the constant system \(K_1(A)\). Apply Theorem 4.1. The initial map is the stated corner. The same argument applies to a Banach completion of the finite matrix algebra whenever its normed corner system has the preceding lesson's hypotheses. \(\square\)

5. Components detected by spectra, winding and index

Theorem 5.1. If \(M\) is any von Neumann algebra, then \(K_1(M)=0\).

Proof. We construct a bounded self-adjoint logarithm using only continuous functional calculus and the strong closure of \(M\subset B(H)\). This gives the bounded argument used in the usual Borel-calculus proof [B06, I.6.2.1–I.6.2.4] without assuming that additional calculus here.

First, any increasing sequence \(0\leq b_k\leq C I\) of operators has a strong limit. For each \(\xi\in H\), the numbers \(\langle b_k\xi,\xi\rangle\) increase to a finite limit. If \(l\geq k\), positivity and \((b_l-b_k)^2\leq C(b_l-b_k)\) give

Thus \(b_k\xi\) is Cauchy. Its limit defines a bounded linear operator \(b\), and passage to limits in inner products shows \(0\leq b=b^*\leq CI\). If all \(b_k\) belong to \(M\), so does \(b\), since a von Neumann algebra is strongly closed. This argument uses no countable basis for \(H\).

For a unitary \(v\), parameterize the circle by \(z=e^{i\theta}\), \(0\leq\theta\leq2\pi\), and define the continuous functions

Both endpoint values are zero, so these are well-defined on the circle. They increase and lie between 0 and \(2\pi\). Continuous functional calculus gives \(b_k=f_k(v)\in M\) with \(0\leq b_k\leq b_{k+1}\leq2\pi I\). Let \(b\in M\) be their strong limit.

We show \(e^{ib}=v\). Put \(x=2\pi-\theta\). If \(f_{2k}(e^{i\theta})=2kx\), then \(f_k=kx\) and

If instead \(f_{2k}(e^{i\theta})=\theta\), the same absolute difference is at most \(\theta-f_k=f_{2k}-f_k\). The absolute difference is always at most 2. Squaring therefore gives the continuous pointwise inequality

Functional calculus preserves positivity, so for every \(\xi\)

Also \(e^{ib_k}\to e^{ib}\) strongly: powers converge strongly under uniform boundedness, and the exponential power series has uniformly small norm tails for \(\|b_k\|,\|b\|\leq2\pi\). Thus \(e^{ib}=v\).

Apply this construction to \(v=-u^*\) and put \(h=\pi I-b\). Then

Indeed \(-\pi I\leq h\leq\pi I\), and \(e^{ih}=-e^{-ib}=-v^*=u\). The path \(\exp(ith)\) joins 1 to \(u\) in norm. Every \(M_n(M)\) is again a von Neumann algebra, so the construction applies at every matrix size. Polar decomposition then joins every invertible matrix to a unitary in the identity component. Proposition 1.2 identifies this ordinary unital computation with (1.2), proving the result. No separability or finiteness hypothesis on \(M\) is used. \(\square\)

Strong closure keeps the constructed logarithm inside the von Neumann algebra. A unitary in an arbitrary C*-algebra need not have a self-adjoint logarithm in that algebra; the circle example supplies the obstruction.

Sphere maps and unitary transport

The circle computation needs the fundamental group of finite unitary groups. We prove the required topology here, together with the second homotopy group that will be used in Suspension, higher K-groups and the long exact sequence. A based homotopy keeps the chosen basepoint fixed throughout. The notation \(\pi_d(Y,y_0)\), for \(d\geq1\), means based homotopy classes of maps \(S^d\to Y\).

Lemma (maps into a higher-dimensional sphere). If \(0\leq d<m\), every continuous based map \(f:S^d\to S^m\) contracts to its basepoint through based maps.

Proof. Realize \(S^d\) as the radial image of the boundary of a \((d+1)\)-simplex containing the origin in its interior. Rotate this realization so that one vertex is the domain basepoint. Radial projection is a homeomorphism: each ray meets the boundary once, continuously in the ray. Barycentric subdivisions give arbitrarily small simplices before radial projection. Indeed a subdivided simplex has vertices that are barycentres of nested faces. If these faces have \(r\) and \(s\) vertices, with \(r\leq s\leq d+1\), their barycentres differ by at most \((1-r/s)\) times the original diameter. Thus each subdivision reduces the mesh by at least the factor \(d/(d+1)\) when \(d\geq1\). Uniform continuity of radial projection and of \(f\) makes the image oscillation on every radial simplex less than a chosen \(\delta<1/4\). For \(d=0\) the two vertices already suffice.

On a simplex with vertices \(v_0,\ldots,v_k\), use its barycentric coordinates \(\lambda_j\) before radial projection and set

These formulas agree on shared faces. At every point \(\|F(x)-f(x)\|<\delta\), so \(F\) never vanishes. The normalized straight interpolation between \(f(x)\) and \(F(x)\) gives a continuous homotopy from \(f\) to \(g\). It fixes every vertex, in particular the basepoint.

The image of \(g\) on each simplex lies in the real linear span of its at most \(d+1\) vertex images. This is a proper subspace of \(\mathbb R^{m+1}\). A finite union of these subspaces misses a unit vector \(q\): choose a nonzero linear functional vanishing on each subspace; along the curve \((1,t,\ldots,t^m)\) each functional is a nonzero polynomial and has only finitely many roots. A parameter outside all those finite sets, followed by normalization, gives \(q\). Since \(g\) includes the basepoint image, \(q\) is different from that image.

The complement \(S^m\setminus\{q\}\) is homeomorphic to \(\mathbb R^m\). Explicitly, after rotating \(q\) to the last coordinate vector, stereographic projection and its inverse are

Straight interpolation in \(\mathbb R^m\) from the stereographic image of \(g\) to the image of its basepoint contracts \(g\) while fixing that basepoint. Combining the two homotopies proves the assertion. In particular \(S^2\) is path connected and every based loop on it contracts; and \(\pi_1(S^m)=\pi_2(S^m)=0\) when \(m>2\). \(\square\)

Lemma (unitary transport of vectors). Let \(X\) be compact Hausdorff and let \(h:X\times[0,1]\to S^{2n-1}\subset\mathbb C^n\) be continuous. There is a continuous \(T:X\times[0,1]\to U(n)\) with

At every \(x\) where \(h(x,t)\) is constant in \(t\), we may take \(T(x,t)=I_n\) throughout.

Proof. First transport two nearby unit vectors \(a,b\). Put \(R_a=aa^*\) and \(R_b=bb^*\). If \(\|a-b\|<1/2\), then \(\|R_a-R_b\|\leq2\|a-b\|<1\). The matrix

is invertible by the Neumann series. Also \(zR_a=R_bz\), and adjointing this equality shows that \(z^*z\) commutes with \(R_a\). Its positive inverse square root does as well. Consequently

is unitary and satisfies \(vR_av^*=R_b\). It depends continuously on \((b,a)\) and equals identity when \(b=a\), by continuous functional calculus. Thus \(va=bc\) for the scalar \(c=b^*va\), which has modulus one. Correcting this phase gives

Both factors are unitary; \(W(b,a)a=b\), and \(W(a,a)=I_n\).

Compactness gives a finite subdivision \(0=t_0<\cdots<t_N=1\) such that \(\|h(x,t)-h(x,t_j)\|<1/2\) for all \(x\) and \(t_j\leq t\leq t_{j+1}\). This uniform subdivision follows by covering the compact parameter product with neighborhoods of uniform small oscillation. Start with \(T(x,0)=I_n\) and, on successive subintervals, define

The formulas agree at the subdivision points because \(W(a,a)=I_n\). Induction proves the transport identity, and a constant vector path makes every factor identity. \(\square\)

The same construction exhibits the bundle \(U(n-1)\to U(n)\to S^{2n-1}\), whose projection is the first column. Given a unit vector \(a\), extend it to an orthonormal basis and let \(F_a\) be the corresponding unitary. For \(\|b-a\|<1/2\), every unitary with first column \(b\) has a unique expression

Indeed \(F_a^*W(b,a)^*U\) fixes the first basis vector and its orthogonal complement. This gives a continuous local trivialization and its inverse. The transport lemma supplies the compact-family homotopy lifting needed below without appealing to a general homotopy exact sequence. These standard sphere and bundle facts are also treated in Hatcher [H, Corollary 4.9 and Example 4.55].

Proposition (the determinant and the second homotopy group). For each \(n\geq1\), determinant induces an isomorphism

The positive generator is \(t\mapsto\operatorname{diag}(e^{2\pi it},1_{n-1})\). Moreover \(\pi_2(U(n),I_n)=0\) and \(\pi_2(GL_n(\mathbb C),I_n)=0\).

Proof. We first check the scalar case. A based loop \(u:[0,1]\to U(1)\), with endpoints 1, has a continuous real argument \(\ell\) such that \(u(t)=e^{2\pi i\ell(t)}\) and \(\ell(0)=0\). Choose arguments on short arcs and join them by integer adjustments, exactly as in Vector bundles and finitely generated projective modules, Lemma 5.2. Then \(\ell(1)=k\in\mathbb Z\), its winding. The based homotopy

joins \(u\) to the loop of winding \(k\). Lemma 5.2 proves that winding is homotopy invariant and additive, so it classifies based scalar loops.

A based map \(f:S^2\to U(1)\) has a continuous real argument as well. To construct it, start with argument 0 at the basepoint, continue it along a path to each point, and take the endpoint argument. Path lifting by successive short arcs gives existence and uniqueness on an interval. Two chosen paths differ by a loop on \(S^2\). The sphere lemma contracts that loop, so its image under \(f\) has winding zero by Lemma 5.2. Thus the endpoint argument is independent of the path. It is continuous: near a point, \(f\) lies in one short arc, and continuation along paths inside a small path-connected neighborhood agrees with that arc's continuous argument plus a fixed integer. Scaling this global argument to zero contracts \(f\) through based maps. Hence \(\pi_2(U(1))=0\).

Now let \(f:S^d\to U(n)\) be based, where \(d=1\) or 2 and \(n\geq2\). Its first column \(h(x,0)=f(x)e_1\) is a based map into \(S^{2n-1}\). Since \(d<2n-1\), the sphere lemma supplies a homotopy \(h(x,t)\) to the constant \(e_1\), keeping the basepoint fixed. Apply unitary transport and put

This is a based unitary homotopy. At its endpoint the first column is \(e_1\), so \(f_1=\operatorname{diag}(1,g)\) for a based \(g:S^d\to U(n-1)\). Repeating reduces the original map to a scalar map in the last coordinate.

For \(d=1\), determinant winding is preserved by all these homotopies. A loop of winding zero reduces to a scalar loop of winding zero, which contracts by the scalar argument. Conversely the displayed positive diagonal loop has winding one. Additivity of winding under concatenation now proves that determinant gives an isomorphism on \(\pi_1(U(n))\). For \(d=2\), every reduced scalar map contracts by the global-argument construction, proving \(\pi_2(U(n))=0\).

Finally the based polar deformation

retracts \(GL_n(\mathbb C)\) onto \(U(n)\) and \(\mathbb C^\times\) onto \(U(1)\), fixing identity. The positive factor in its determinant has winding zero. It therefore transfers both assertions to the general linear groups and proves (5.2). \(\square\)

The same proof also contracts based loops and two-sphere maps into \(SU(n)\). Given the unitary null-homotopy \(F\), replace it by \(\operatorname{diag}((\det F)^{-1},1_{n-1})F\). Its initial map is unchanged, its determinant is constantly 1, and it preserves the basepoint and the final identity.

Winding detects the circle group

Theorem 5.2 (the circle). With counterclockwise winding number,

In particular \(D([z^k])=k\), for every \(k\in\mathbb Z\).

Proof. By Proposition 1.2 use ordinary matrices \(g:S^1\to GL_n(\mathbb C)\). Norm paths between these functions are exactly jointly continuous homotopies: compactness of \(S^1\) makes a continuous homotopy uniformly continuous in its path parameter. Determinant preserves homotopies and satisfies

Winding number is homotopy invariant and additive under multiplication, by the winding-number prerequisite. Thus (5.3) is a well-defined group homomorphism.

To check injectivity at every finite size, fix the basepoint \(1\in S^1\) and multiply \(g\) on the right by \(g(1)^{-1}\). This does not change its homotopy class, since the constant matrix \(g(1)^{-1}\) is connected to 1 in \(GL_n(\mathbb C)\). It does not change determinant winding either. The resulting map is a based loop. If its determinant has winding zero, (5.2) makes the based loop nullhomotopic. Hence the original matrix function lies in the component of the identity. Every stable class of winding zero is therefore zero. Finally the scalar functions \(z^k\) realize all integers, which proves surjectivity. The generator orientation is the one in (5.2). \(\square\)

Example 5.3 (an interval and the real line). The homomorphisms

give a pointwise norm-continuous homotopy from constant evaluation at 0 to the identity. Evaluation factors through \(\mathbb C\), whose \(K_1\) is zero. Homotopy invariance gives \(K_1(C([0,1]))=0\).

In contrast, the one-point compactification identifies \(C_0(\mathbb R)^+\) with \(C(S^1)\). Proposition 1.2 and Theorem 5.2 give

A normalized unitary generator is

Indeed \(u-1\in C_0(\mathbb R)\), \(u(\infty)=1\) and \(|u(t)|=1\). Parameterize the compactified line by \(t=-\cot(\pi s)\), \(0<s<1\), with both endpoints sent to infinity. Direct substitution gives \(u(t)=e^{2\pi is}\), so (5.5) has winding \(+1\) with this orientation. Its \(k\)-th powers represent every integer, including negative integers. Contractibility of the underlying space \(\mathbb R\) does not make its function algebra contractible through homomorphisms in the required norm topology.

Theorem 5.4 (the Calkin algebra, with sign fixed). For every infinite-dimensional Hilbert space \(H\), separable or not, put \(Q(H)=B(H)/\mathcal K(H)\). Then

For a matrix representative, \(T\) is a lift acting on \(H^n\). If \(H\) is finite-dimensional, \(Q(H)=0\) and \(K_1(Q(H))=0\).

Proof. A matrix is compact on \(H^n\) exactly when every entry is compact on \(H\): compression by coordinate maps proves necessity, and the finite sum of compact coordinate operators proves sufficiency. Consequently

By the Fredholm prerequisite, Theorem 1.1, every invertible quotient representative lifts to a Fredholm operator. Any two lifts differ compactly, so Corollary 3.3 makes their indices equal. Its Theorem 2.1 proves additivity under multiplication; Theorem 3.2 proves homotopy invariance; and Theorem 4.2 says that the kernel of this component homomorphism is precisely the identity component. These statements apply to \(H^n\) for every \(n\).

For completeness, surjectivity does not need separability of \(H\). Choose a countably infinite orthonormal sequence, let \(H_0\) be its closed span and let \(S\) be the unilateral shift on \(H_0\). On \(H=H_0\oplus H_0^\perp\) take

This is an isometry with zero kernel and one-dimensional cokernel, and \(1-TT^*\) has rank one. Thus \(q(T)\) is unitary and \(\kappa(T)=1\). Its powers and adjoint powers realize all integers. The index is therefore an isomorphism on the finite component group in each size. Adding an identity block to a lift leaves both defects, and hence the index, unchanged. Passage to stable components gives (5.6). \(\square\)

The sign in (5.6) follows the existing Fredholm lesson: cokernel dimension minus kernel dimension, so the shift has value \(+1\). With the also common convention

the same shift has value \(-1\). The next lesson defines an index connecting map; its sign will be compared to (5.8) explicitly. An abstract isomorphism with \(\mathbb Z\) alone does not fix this choice.

At a general C*-algebra the map from components of a single \(GL_n(A)\) to \(K_1(A)\) need not be an isomorphism. The circle and Calkin proofs establish that it is an isomorphism in those particular examples. All general definitions and proofs above retain arbitrary finite stabilization.

6. A determinant quotient with a canonical section

Let \(X\) be compact Hausdorff. Give \([X,S^1]\), the set of homotopy classes of continuous maps to the circle, the abelian group operation induced by pointwise multiplication. Radial deformation \(z\mapsto z/|z|\) identifies it with \([X,\mathbb C^\times]\).

Proposition 6.1. Determinant induces a split surjection

Its section sends a circle-valued function \(g\) to its scalar matrix class \([g]\). In particular

Proof. Represent classes using ordinary invertible matrix functions, by Proposition 1.2. Determinant preserves homotopy and identity stabilization. The block-sum identity for determinant proves that it is a group homomorphism to \([X,\mathbb C^\times]\), hence to \([X,S^1]\). Compactness of \(X\) identifies joint continuous homotopies with norm paths, so this works for arbitrary compact Hausdorff \(X\), without a metrizability assumption.

The scalar inclusion is well defined on homotopy classes. It is a homomorphism because Theorem 2.2 gives \([gh]=[g]+[h]\) for scalar functions. Determinant of a scalar matrix is that function, so \(D_X([g])=[g]\) as a homotopy class of maps. The composite of determinant with scalar inclusion is the identity. If \(s\) denotes this section, the explicit splitting sends \(x\) to \((x-s(D_Xx),D_Xx)\), with inverse \((y,a)\mapsto y+s(a)\). This proves (6.2). \(\square\)

For the circle, Theorem 5.2 says that the kernel in (6.2) vanishes. For other spaces a determinant may leave a nonzero kernel. The homotopy quotient in this assertion is essential, as the comparison with algebraic K-theory below illustrates. Proposition 6.1 asserts a direct summand and does not assert that every stable class has a scalar representative.

Algebraic and topological \(K_1\)

The determinant quotient above uses homotopy classes of functions. It must be distinguished from algebraic K-theory, where no path is imposed. For a unital C*-algebra \(A\), considered just as a ring, write \(G=GL_\infty(A)\) and define

This is the abelianization of the stabilized algebraic group, as in [Zois, Lecture 17]. The topological group in this lesson is instead its quotient by finite stabilized norm paths to the identity, by Proposition 1.2.

Proposition 6.2. Sending an invertible to its topological class gives a surjective homomorphism

For the complex numbers this map is \(\mathbb C^\times\to0\).

Proof. Theorem 2.2 proves that the topological quotient is abelian, so every commutator maps to zero. Hence the quotient map factors through (6.3), and it remains surjective because every topological class has a finite invertible representative.

We verify the complex algebraic computation explicitly. Determinant on \(GL_\infty(\mathbb C)\) is onto \(\mathbb C^\times\), and its kernel is \(SL_\infty(\mathbb C)\). Every determinant-one matrix is a product of elementary matrices \(E_{ij}(a)=1+ae_{ij}\). To see this, Gaussian elimination uses row additions and determinant-one row exchanges to reach a diagonal matrix; a nonzero entry in a pivot column can be moved into the pivot position by the two-row matrix \(w(1)\) below. Clearing successive columns gives a nonzero diagonal, whose entries have product one. For a nonzero scalar \(a\), diagonal two-row factors are elementary because

A diagonal determinant-one matrix is a product of these factors acting on each coordinate and the last coordinate. Reversing the elimination expresses the original matrix as a product of elementary matrices. For three distinct indices, direct multiplication gives

After stabilization a third index is always available. Thus every elementary generator, and hence the whole determinant kernel, is in the commutator subgroup. Conversely determinants of commutators are one. This proves \(K_1^{\mathrm{alg}}(\mathbb C)=\mathbb C^\times\). Polar deformation joins an invertible complex matrix to a unitary one, and diagonalizing that unitary joins it to the identity by rotating its finitely many eigenvalues. Therefore \(K_1(\mathbb C)=0\), as already computed in §5, and (6.4) kills the entire algebraic group. \(\square\)

For example, the scalar \(2\) is nontrivial in the algebraic group, detected by its determinant. The invertible path \(t\mapsto2-t\) makes its topological class zero. Algebraic \(K_1\) remembers a value; topological \(K_1\) identifies values joined by an invertible path. On \(C(S^1)\), winding survives those paths, which is why it remains in our circle computation.

7. Exercises with complete solutions

Exercise 7.1 — The rotation and its inverse (basic). Write the Whitehead homotopy explicitly, check invertibility along it and verify the scalar part when \(A\) is nonunital.

Solution. With \(c=\cos\theta\), \(s=\sin\theta\), multiplication in (2.1) gives

All scalar identities here are \(1_n\), and the order of the factor \(u\) is retained. The factorization in (2.1) supplies the inverse

since \(R_{-\theta}=R_\theta^{-1}\). Multiplication in either order cancels the consecutive factors to \(1_{2n}\). Thus every point is invertible, even when \(u,v\) do not commute. The endpoint values are \(\operatorname{diag}(uv,1_n)\) and \(\operatorname{diag}(u,v)\). If the scalar parts of \(u,v\) are 1, (7.1) has scalar diagonal blocks \(c^2+s^2=1\) and scalar off-diagonal blocks zero. It lies in \(G_{2n}^1(A)\) throughout. If \(u,v\) are unitary, the factorization is a product of unitaries. Setting \(v=u^{-1}\) gives a path from the identity to \(u\oplus u^{-1}\); reversing it contracts that stabilized representative.

Exercise 7.2 — Direct sums and their maps (basic). Show that \(K_1(A\oplus B)\cong K_1(A)\oplus K_1(B)\), and identify both directions of the isomorphism.

Solution. Let \(p_A,p_B\) be the two projections and \(i_A,i_B\) the inclusions. The forward map is \(((p_A)_*,(p_B)_*)\). Given normalized representatives \(1+a\) over \(A\) and \(1+b\) over \(B\), enlarge them by identity blocks to a common size and take \(1+(a,b)\). Its inverse has entries reconstructed from the two normalized inverses, as in Proposition 3.2. Coordinatewise homotopies show that its class depends only on the two given classes.

This inverse is also \((x,y)\mapsto(i_A)_*x+(i_B)_*y\): the product of the normalized matrices \(1+(a,0)\) and \(1+(0,b)\) is \(1+(a,b)\), because the two ideals have zero cross products. Applying the projections returns \((x,y)\). Conversely the reconstructed pair is the original normalized matrix over the direct sum. These are inverse homomorphisms, and the argument covers nonunital algebras.

Exercise 7.3 — Contractible algebras and the real-line generator (intermediate). Show that a contractible C*-algebra has zero \(K_1\), and compute \(K_1(C_0(\mathbb R))\).

Solution. Contractible here means that the identity homomorphism of the algebra is homotopic to the zero homomorphism through *-homomorphisms, pointwise continuously in norm. Homotopy invariance gives \(\operatorname{id}_{K_1(A)}=0_*\). The zero homomorphism sends every normalized \(1+a\) to an identity matrix, so \(0_*=0\). A group whose identity map is zero is the zero group.

For example the cone \(C_0((0,1],A)\), viewed as continuous functions on \([0,1]\) vanishing at 0, is contracted by \(f(s)\mapsto f(ts)\), \(0\leq t\leq1\). Uniform continuity gives pointwise norm continuity of this homotopy. This supplies an actual algebra homotopy, rather than merely a contraction of an underlying vector space.

For \(C_0(\mathbb R)\), the external unitization is \(C(S^1)\). Hence (5.4) gives \(K_1\cong\mathbb Z\), with normalized generator (5.5). The substitution \(t=-\cot(\pi s)\) verifies winding \(+1\) directly. Thus this algebra is not contractible through homomorphisms. The space \(\mathbb R\) itself is contractible, but that fact supplies no norm-continuous contraction of the identity homomorphism of \(C_0(\mathbb R)\) to zero.

Exercise 7.4 — \(B(H)\) without a dimension restriction (intermediate). Prove \(K_1(B(H))=0\) using a self-adjoint logarithm of every unitary.

Solution. Theorem 5.1 constructs for every unitary in \(B(H)\) a self-adjoint \(h\) with \(\|h\|\leq\pi\) and \(u=\exp(ih)\). It uses the increasing continuous functions \(f_k\) applied to \(-u^*\), their strong limit \(b\), and \(h=\pi I-b\). Apply that same construction on \(H^n\), since \(M_n(B(H))\cong B(H^n)\). The norm path \(t\mapsto\exp(ith)\) connects every matrix unitary to 1. The polar-component theorem connects every invertible matrix to a unitary, so every matrix component class is zero. The stable group is zero as well. This includes finite-dimensional and nonseparable Hilbert spaces. If \(H=0\), the algebra is zero and the normalized definition gives the same conclusion immediately.

Exercise 7.5 — Two independent torus windings (advanced). For \(X=\mathbb T^2\), construct independent homomorphisms from \(K_1(C(X))\) to \(\mathbb Z\) using the two circle factors, and prove that the coordinate classes generate a \(\mathbb Z^2\) direct summand.

Solution. Orient the circles counterclockwise and use the loops \(\ell_1(z)=(z,1)\), \(\ell_2(z)=(1,z)\). For a matrix representative put

Restriction is a homomorphism of function algebras, and the circle computation proves that each \(w_j\) is a well-defined homomorphism on stable classes. Alternatively determinant homotopy invariance, identity stabilization and block multiplicativity check this directly. Let \(z_1,z_2\) be the coordinate functions. Their winding vectors are

The map

is a homomorphism by Theorem 2.2 and satisfies \((w_1,w_2)\sigma=\operatorname{id}_{\mathbb Z^2}\). Therefore \(\sigma\) is injective, both winding maps are independent, and

The decomposition follows explicitly by subtracting \(\sigma((w_1,w_2)(x))\) from any class \(x\). This solves the direct-summand question without assuming that the displayed subgroup is the entire torus \(K_1\)-group.

References and prerequisite locators

[B98] B. Blackadar, K-Theory for Operator Algebras, second edition, Cambridge University Press, 1998, §3.4 (invertible components), §8.1 (definition, Whitehead lemma and properties of \(K_1\)), and Corollary 8.3.7 (unitization). Proposition 1.2 gives the unitization result directly, so it does not depend on that book's later exact-sequence argument. Author's corrected second edition.

[B06] B. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras, Springer, 2006, revised author version 2017, V.1.2.1–V.1.2.7 and V.1.2.17; I.6.2.1–I.6.2.4 for bounded Borel calculus of a normal operator. The boundary-map convention in V.1.2.13 will be compared with (5.8) in the next lesson.

[E24] H. Emerson, An Introduction to C*-Algebras and Noncommutative Geometry, Birkhäuser, 2024, §8.3. That text begins with suspension groups. Here the stable-invertible definition comes first; its identification with the suspension description is proved in the suspension lesson.

[H] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002, Corollary 4.9, printed p. 349, and §4.2, Example 4.55, printed p. 383, for the standard sphere contractions and unitary-group bundles. Section 5 gives the complete sphere approximation, compact-family transport and low-dimensional group proofs used in this course. Official author PDF.

The prerequisite Invertible components and exponential laws, in Foundations of von Neumann algebras: remaining topics, supplies Recall 1.1 and Theorem 4.1; its Theorem 5.2 gives a compatible winding convention. The proof actually used for winding is the preceding bundle lesson's Lemma 5.2. The exponential-law lesson's Theorems 2.1 and 3.1 concern one-parameter groups and product formulas; the component facts needed here are in Recall 1.1.

The prerequisite Fredholm operators and the stable index, in the same remaining-topics course, supplies Theorems 1.1, 2.1, 3.2 and 4.2, Corollary 3.3, and Lemma 4.1. Its index is \(\kappa\), as specified in (5.6); the extension of its shift argument to arbitrary infinite-dimensional \(H\) is included above.

• [Zois] I. P. Zois, 18 Lectures on K-Theory, arXiv:1008.1346v1, Lecture 17, “The Whitehead and the Steinberg Groups,” printed pp. 99–102. Freely available source. Proposition 6.2 uses the algebraic/topological distinction and supplies its own elimination and commutator calculations. No source expression is adapted.