# Projection comparison and normal center-valued traces
*Written by GPT-6.1 Sol (OpenAI), Ultra, October 2026. New original text and embedded diagrams: public domain (CC0).*
Projection comparison lets finite corners be matched without a dimension function. We use it first to prove finite matrix stability and finite projection sums. Abelian corners and coherent dyadic partitions then provide monic projections; these allow normal almost traces to converge in norm to the center-valued trace. Every orthogonal sum below is the strong net of finite partial sums.
Let \(M\subseteq B(H)\) be a unital strongly closed self-adjoint algebra on an arbitrary Hilbert space. The unit is denoted by \(1\); when working in a corner its unit is the corner projection. The zero Hilbert space and zero corner are allowed and all projection assertions there are immediate. A projection is an operator \(p=p^*=p^2\). Write \(p\le q\) when \(pH\subseteq qH\), equivalently \(pq=qp=p\). Put
\[
p\sim q \iff \exists v\in M:\ v^*v=p,\ vv^*=q,
\qquad
p\precsim q \iff \exists v\in M:\ v^*v=p,\ vv^*\le q.
\]
Such an operator \(v\) is a partial isometry. A projection \(p\) is **finite** if every \(v\in M\) with \(v^*v=p\) and \(vv^*\le p\) has \(vv^*=p\). Thus \(M\) is finite exactly when its unit is finite, or equivalently every isometry in \(M\) is unitary.
The external proof inputs are H00 (Hilbert completeness, adjoints, bounded operators), H01 (orthogonal projections and strong closure), H02 (positive supports), the bounded polar construction T04a, and the C*-algebra completeness/calculus/order arguments F01–F08. Zorn's lemma is used for maximal orthonormal sets and for maximal matching families. No trace, countable decomposition, type classification, center-valued trace, predual realization, automatic normality, or properly infinite halving theorem is used. H03 is available but is not needed for this chain.
For an arbitrary Hilbert space, the coordinate version of H00 causes no restriction: a maximal orthonormal set exists by Zorn. A nonzero vector perpendicular to its closed span would enlarge it, so its span is dense. Finite orthogonal projections and completeness identify the space isometrically onto the square-summable coordinates on that set. Conversely square-summable coordinates have norm-convergent finite partial sums. This includes arbitrary cardinalities and the empty basis of the zero space.
*Figure 1. The exact comparison, complement and matrix-corner route to finite orthogonal sums. The column operator is computed in C10; this is an operator schematic and makes no assertion about finite Hilbert-space dimension. Human proof methods: Peterson, Theorem 5.1.10 and Propositions 5.2.7–5.2.8.*
## C01. Strong multiplication, supports and corners
If \(A_i\to A\) and \(B_i\to B\) strongly and \(\sup_i\|A_i\|\le C\), then
\[
\|(A_iB_i-AB)\xi\|
\le C\|(B_i-B)\xi\|+\|(A_i-A)B\xi\|\longrightarrow0.
\]
This is the only strong-product passage used below. Strong convergence of both an operator and its adjoint permits applying this estimate to their products. Norm closure of \(M\) follows from strong closure, so its C*-calculus is available by H00 and F01–F08.
H02 supplies the support \(s(h)\in M\) of a positive \(h\in M\), the projection onto \(\overline{hH}\). For \(x\in M\), T04a supplies
\[
x=v|x|,\qquad |x|=(x^*x)^{1/2},\qquad
v^*v=[\overline{|x|H}],\quad vv^*=[\overline{xH}],\quad v\in M.
\]
Its membership proof is the uniformly bounded strong limit
\(x(|x|+\varepsilon1)^{-1}\to v\). Also
\(\ker |x|=\ker x\), because \(\||x|\xi\|=\|x\xi\|\).
If \(v^*v=p\), then \(v\) vanishes on \((1-p)H\) and is an isometry on \(pH\); its range there is closed. Its range projection is \(vv^*\), and \(v=vp=(vv^*)v\). These facts follow by evaluating \(\|v\xi\|^2=\langle\xi,p\xi\rangle\) and by Hilbert completeness. In particular, if both supports are at most \(e\), then \(v= eve\).
For every projection \(e\in M\), the corner \(eMe\) on \(eH\) is a unital strongly closed self-adjoint algebra with unit \(e\): extend its strong limits by zero on \((1-e)H\) and use strong closure of \(M\). The same applies to \(M_n(M)\) on \(H^{\oplus n}\): a strong operator limit has strongly convergent entries, obtained by coordinate inclusions and projections, so every limiting entry belongs to \(M\). Finite matrices are bounded by the triangle inequality, and H00 gives their C*-norm and completeness. A projection \(e\) is finite in \(M\) exactly when the unit of \(eMe\) is finite, by the support identity above.
For later optional H03 topology statements, the commutant definition also follows directly from these Hilbert inputs. If \(T\in M''\) and \(\xi_1,\ldots,\xi_n\in H\), let \(K\) be the closure of \(\{(x\xi_1,\ldots,x\xi_n):x\in M\}\) in \(H^{\oplus n}\). This set is already a linear subspace because \(M\) is linear. It reduces every diagonal operator from \(M\), so its orthogonal projection has entries in \(M'\). The diagonal operator from \(T\) commutes with those entries and hence with this projection. The tuple \((\xi_1,\ldots,\xi_n)\) belongs to \(K\), since \(1\in M\); therefore its image under \(T\) belongs to \(K\). Finite-vector approximation follows: for any positive error some \(x\in M\) approximates \(T\) on all these vectors. Choosing such \(x\) over finite sets of vectors and decreasing errors constructs a net converging strongly to \(T\). Strong closure gives \(T\in M\). Thus \(M=M''\), proved here without importing a bicommutant or density theorem. Corners, matrices and the center are strongly closed self-adjoint algebras too, so this argument applies to them. H03 consequently supplies their concrete preduals whenever used in the extension.
## C02. Arbitrary projection lattice and orthogonal sums
For finitely many projections \(p_1,\ldots,p_m\), their join is
\(s(p_1+\cdots+p_m)\). Indeed a vector is in the kernel of this positive sum exactly when it is in every \(\ker p_i\), because its quadratic value is \(\sum_i\|p_i\xi\|^2\). Thus the support has range \(\overline{p_1H+\cdots+p_mH}\).
For a set-indexed family \((p_i)_{i\in I}\), let \(p_F\) be the join for a finite \(F\subset I\), with \(p_\varnothing=0\). These projections converge strongly to the projection \(P\) onto
\[
L=\overline{\operatorname{span}\bigcup_{i\in I}p_iH}.
\]
They vanish on \(L^\perp\). Every vector in a finite sum of the ranges is fixed eventually; density in \(L\) and the common norm bound one prove convergence on all of \(H\). Strong closure puts \(P\) in \(M\), and its range description makes it the least upper bound \(\bigvee_i p_i\). Meets are
\(\bigwedge_i p_i=1-\bigvee_i(1-p_i)\). Empty joins and meets are respectively zero and one. In particular a meet has range equal to the intersection of the ranges.
If the \(p_i\) are pairwise orthogonal, their finite sums are their finite joins, so
\(\sum_i p_i:=\operatorname{s\!-!lim}_F\sum_{i\in F}p_i=\bigvee_i p_i\).
Every arbitrary sum in this note means this net of finite partial sums, rather than a chosen enumeration.
Now suppose \(v_i\in M\) have pairwise orthogonal initial projections \(p_i=v_i^*v_i\) and pairwise orthogonal final projections \(q_i=v_iv_i^*\). For distinct indices,
\[
v_i^*v_j=v_i^*q_iq_jv_j=0,
\qquad v_iv_j^*=v_ip_ip_jv_j^*=0.
\]
For finite \(F\), \(v_F=\sum_{i\in F}v_i\) therefore satisfies
\[
v_F^*v_F=\sum_{i\in F}p_i,\quad v_Fv_F^*=\sum_{i\in F}q_i,
\qquad \|v_F\|\le1.
\]
For each \(\xi\), the finite sums \(\sum_{i\in F}\|v_i\xi\|^2\) are bounded by \(\|\xi\|^2\). Choose a finite \(F_0\) whose sum is within \(\varepsilon^2\) of their supremum. For finite \(F,G\supseteq F_0\), orthogonality gives
\(\|(v_F-v_G)\xi\|^2\le\sum_{i\notin F_0}\|v_i\xi\|^2\le\varepsilon^2\), where the tail is the supremum of finite tail sums. Consequently \(v_F\) is strong Cauchy; completeness constructs its bounded strong limit \(v\). The same argument for adjoints constructs a strong limit \(w\) of \(v_F^*\). Matrix coefficients give \(w=v^*\). C01 then gives
\[
v^*v=\sum_i p_i,\qquad vv^*=\sum_i q_i,\qquad v\in M.
\tag{C02.1}
\]
This includes empty families, whose sum is zero. It proves both arbitrary orthogonal-sum equivalence and orthogonal additivity of subequivalence: if \(p_i\precsim q_i\) and the two projection families are separately orthogonal, choose implementing partial isometries with final supports \(r_i\le q_i\) and apply (C02.1).
## C03. Elementary equivalence and finiteness facts
Equivalence is reflexive (implemented by \(p\)), symmetric (use the adjoint), and transitive: if \(v\) implements \(p\sim q\) and \(w\) implements \(q\sim r\), then \(wv\) implements \(p\sim r\). Subequivalence is transitive by the same product: when \(vv^*\le q=w^*w\), its final support is \(wvv^*w^*\le ww^*\).
For \(r\le p=v^*v\), the restriction \(vr\) implements
\(r\sim vrv^*\le vv^*\). Thus equivalence preserves subprojection comparisons and gives an algebraic *-isomorphism of the two corners, with maps \(a\mapsto vav^*\) and \(b\mapsto v^*bv\).
Finiteness is hereditary. If \(r\le p\) and \(u^*u=r\), \(uu^*\le r\), then \(u+(p-r)\) has initial support \(p\) and final support \(uu^*+(p-r)\le p\); all mixed products vanish because the two initial and final supports are orthogonal. Finiteness of \(p\) forces \(uu^*=r\). Finiteness is also invariant under equivalence, since the displayed corner maps transport an isometry with a proper final support to one in the other corner. It follows that
\[
p\precsim q\text{ and }q\text{ finite}\quad\Longrightarrow\quad p\text{ finite}.
\tag{C03.1}
\]
The zero projection is finite.
## C04. Central support and contact between corners
Every element of \(M\) is a linear combination of unitaries. For a self-adjoint contraction \(h\), F06–F08 give a commuting square root \(k=(1-h^2)^{1/2}\); \(h+ik\) is unitary and \(h\) is its real part. Scale arbitrary self-adjoint elements and decompose an arbitrary element into its real and imaginary parts.
For \(p\in\mathcal P(M)\), define
\[
c(p)=\bigvee_{u\in\mathcal U(M)}upu^*.
\]
It belongs to \(M\) by C02. Conjugation by any unitary permutes the ranges defining its join; hence it fixes \(c(p)\). The preceding unitary span makes \(c(p)\) central. It majorizes \(p\), and any central projection majorizing \(p\) majorizes every conjugate and therefore this join. Thus it is the least central projection majorizing \(p\). Its range is
\[
c(p)H=\overline{\operatorname{span}\{xp\xi:x\in M,\ \xi\in H\}},
\tag{C04.1}
\]
because every \(x\) is a finite linear combination of unitaries. This is the precise meaning of \(c(p)=[MpH]\). It includes \(c(0)=0\).
Equivalent projections have the same central support: a central \(z\) with \(zp=p\) also fixes \(q=vpv^*\), and the adjoint argument reverses the implication.
For arbitrary projections,
\[
c(p)c(q)=0\quad\Longleftrightarrow\quad pMq=\{0\}.
\tag{C04.2}
\]
If the central supports are orthogonal, \(pxq=c(p)c(q)pxq=0\). Conversely, if \(pMq=0\), then
\(\langle xp\xi,yq\eta\rangle=\langle\xi,px^*yq\eta\rangle=0\) for all \(x,y,\xi,\eta\). The two closed spans in (C04.1) are orthogonal, proving the forward product is zero.
Moreover \(pMq\ne0\) exactly when \(p\) and \(q\) have nonzero equivalent subprojections. For a nonzero \(x\in pMq\), C01's polar isometry has initial support under \(q\) and nonzero final support under \(p\); reverse it to match in the stated order. Conversely a matching \(v\) from \(p_0\le p\) onto \(q_0\le q\) has nonzero adjoint \(v^*=pv^*q\in pMq\).
## C05. General comparison and projection Cantor–Bernstein
Consider sets of triples \((p_i,q_i,v_i)\) with nonzero pairwise orthogonal \(p_i\le p\), nonzero pairwise orthogonal \(q_i\le q\), and \(v_i^*v_i=p_i\), \(v_iv_i^*=q_i\). These sets lie in the fixed set \(\mathcal P(M)\times\mathcal P(M)\times M\) and are ordered by inclusion. The empty set is admissible. A union of a chain is admissible, since every two of its members lie together in one member of that chain. Zorn supplies a maximal matching family. Put
\[
P=\sum_i p_i,\quad Q=\sum_i q_i,\quad v=\sum_i v_i,\qquad
p_0=p-P,\quad q_0=q-Q.
\]
C02 proves all sums exist in \(M\) and \(P\sim Q\) via \(v\). If \(p_0Mq_0\ne0\), C04 supplies a further nonzero matching in these residual projections, contradicting maximality. Hence \(c(p_0)c(q_0)=0\). Take \(z=c(q_0)\). Then \(zp_0=0\), \((1-z)q_0=0\), and central restriction of \(v\) gives
\[
zp=zP\sim zQ\le zq,
\qquad
(1-z)q=(1-z)Q\sim(1-z)P\le(1-z)p.
\tag{C05.1}
\]
Thus, for every pair, there is a central projection \(z\) with \(zp\precsim zq\) and \((1-z)q\precsim(1-z)p\). No countability assumption enters the matching. In a factor \(z\) is zero or one, so any two projections are comparable. Zero projections and empty maximal families are already included.
For completeness, two-sided subequivalence gives equivalence without a finiteness assumption. Suppose \(u^*u=p\), \(uu^*\le q\), \(v^*v=q\), \(vv^*\le p\). Put \(p'=vv^*\), \(p_0=p-p'\), and \(w=vu\). This is an isometry on \(pH\) whose range is under \(p'\). Define
\[
p_n=w^np_0(w^*)^n\quad(n\ge0),\qquad e=\sum_{n\ge0}p_n,\qquad f=ueu^*.
\]
The \(p_n\) are pairwise orthogonal: after cancelling common powers of the isometry, \(p_0\) is perpendicular to the range of every positive power of \(w\), because those ranges are under \(p'\). C02 gives \(e\). Strong multiplication gives
\[
vfv^*=wew^*=e-p_0,
\qquad
v(q-f)v^*=p'-(e-p_0)=p-e.
\]
Therefore \(ue\) matches \(e\) onto \(f\), while \(v^*(p-e)\) matches \(p-e\) onto \(q-f\). Their orthogonal sum implements \(p\sim q\). This also covers \(p_0=0\), when the first sum is zero. Together with (C05.1), it gives the usual factor trichotomy: either equivalence, or strict subequivalence in exactly one direction, where strict means subequivalent but not equivalent.
## C06. Complement cancellation inside a finite algebra
Assume now \(M\) is finite, and let \(p\sim q\) via \(v\). Apply C05 to \(1-p\) and \(1-q\). For a central \(z\), choose \(w_1\) with
\[
w_1^*w_1=z(1-p),\qquad w_1w_1^*=r\le z(1-q),
\]
and \(w_2\) with
\[
w_2^*w_2=(1-z)(1-q),\qquad
w_2w_2^*=s\le(1-z)(1-p).
\]
The orthogonal sum \(vz+w_1\) has initial support \(z\) and final support \(zq+r\le z\). Since \(z\le1\) is finite by C03, its final support equals \(z\), forcing \(r=z(1-q)\). Similarly \(v^*(1-z)+w_2\) has initial support \(1-z\), so finiteness forces \(s=(1-z)(1-p)\). Thus
\[
w=w_1+w_2^*,\qquad w^*w=1-p,\quad ww^*=1-q.
\]
Finally \(v+w\) is unitary, and \((v+w)p(v+w)^*=q\). This proves equivalent complements in every finite algebra. It uses finiteness of central subprojections of the already finite unit; it does not use closure of finite projections under sums or joins.
## C07. Dense invertibles in a finite algebra
Let \(N\) be any finite concrete algebra as above, with unit \(e\). For \(x\in N\), write \(x=v|x|\) by C01. The two supports of \(v\) are equivalent. C06 extends \(v\) to a unitary \(u\in N\) by matching their complements. Its added part vanishes on the support of \(|x|\), so \(x=u|x|\). For \(\varepsilon>0\),
\[
x_\varepsilon=u(|x|+\varepsilon e)
\]
is invertible, with inverse \((|x|+\varepsilon e)^{-1}u^*\), and
\(\|x_\varepsilon-x\|\le\varepsilon\). The inverse exists by H02 or the continuous reciprocal on the nonnegative spectrum. Hence invertibles are norm dense in \(N\). This includes each nonzero finite corner; zero corners are handled directly.
## C08. Finite matrix algebras over a finite algebra
If invertibles are norm dense in a unital Banach algebra \(N\), they are norm dense in every \(M_n(N)\). Here are the needed algebra and estimates. The assertion for \(n=1\) is the hypothesis. For \(n>1\), write
\[
A=\begin{pmatrix}a&r\\ c&D\end{pmatrix},
\]
where \(D\in M_{n-1}(N)\), \(r\) is a row and \(c\) a column. Given \(\varepsilon>0\), choose invertible \(b\in N\) with \(\|b-a\|<\varepsilon/3\). By induction choose invertible \(E\in M_{n-1}(N)\) satisfying
\(\|E-(D-cb^{-1}r)\|<\varepsilon/3\). Then
\[
A'=\begin{pmatrix}b&r\\c&E+cb^{-1}r\end{pmatrix}
=\begin{pmatrix}1&0\\cb^{-1}&1\end{pmatrix}
\begin{pmatrix}b&0\\0&E\end{pmatrix}
\begin{pmatrix}1&b^{-1}r\\0&1\end{pmatrix}
\]
is invertible: the triangular factors are inverted by negating their off-diagonal blocks, and the diagonal factor by inverting its blocks. Also \(\|A'-A\|<2\varepsilon/3<\varepsilon\), by the coordinate-block norm bounds. No bound on \(\|b^{-1}\|\) is required; it is fixed before approximating the Schur complement.
A concrete unital C*-algebra with dense invertibles is finite. Indeed, suppose \(V^*V=e\), choose invertible \(A\) with \(\|A-V\|<1\), and observe
\(\|V^*A-e\|<1\). The element \(V^*A\) is invertible: for \(y=e-V^*A\), the series \(\sum_{k\ge0}y^k\) converges by completeness, since its tail is bounded by a geometric tail, and multiplying its partial sums gives an inverse in the limit. Thus \(V^*=(V^*A)A^{-1}\) is invertible. The identity \(V^*V=e\) implies \(V=(V^*)^{-1}\), so \(VV^*=e\).
Applying this to C07 proves
\[
N\text{ finite}\quad\Longrightarrow\quad M_n(N)\text{ finite for every finite }n.
\tag{C08.1}
\]
This is an actual proof of matrix finiteness, rather than a stable-finiteness import. In particular, for a finite projection \(b\in M\), \(\operatorname{diag}(b,b)\) is finite in \(M_2(M)\): its corner is exactly \(M_2(bMb)\), which is finite by (C08.1).
## C09. Central patching of finite projections
Let \((p_i)_{i\in I}\) be finite projections with pairwise orthogonal central supports \(z_i=c(p_i)\). Then \(p=\sum_i p_i\) is finite, even if \(I\) is uncountable. Indeed \(z_ip=p_i\). For \(t^*t=p\) and \(tt^*\le p\), centrality gives
\[
(tz_i)^*(tz_i)=p_i,\qquad (tz_i)(tz_i)^*=z_itt^*\le p_i.
\]
Finiteness of \(p_i\) forces \(z_itt^*=p_i\) for every \(i\). Put \(Z=\sum_i z_i\). Since \(p\le Z\) and \(t=ptp\), the range of \(tt^*\) is under \(Z\). Taking the strong limit of the finite sums \(\sum_{i\in F}z_itt^*\) yields \(tt^*=p\), by C01. This includes the empty family.
The same proof applies to projections supported on any prescribed pairwise orthogonal central projections, without requiring those central projections to be their exact central supports. In particular, for a central \(z\), if \(ze\) and \((1-z)e\) are finite, then \(e\) is finite.
The bounded central assembly needed for the homogeneous components also has a direct proof. If \(z_i\) are pairwise orthogonal central projections and \(x_i\in Mz_i\) with \(\sup_i\|x_i\|\le C\), then finite sums \(x_F=\sum_{i\in F}x_i\) have norm at most \(C\), because
\(\|x_F\xi\|^2=\sum_{i\in F}\|x_i z_i\xi\|^2\le C^2\sum_{i\in F}\|z_i\xi\|^2\).
The tail of the last orthogonal square sum tends to zero, so these sums converge strongly to \(x\in M\); the same holds for their adjoints. One has \(z_i x=x_i\) and \(x=(\sum_i z_i)x\), and the preceding bound and restriction to each central summand give
\[
\|x\|=\sup_i\|x_i\|.
\tag{C09.1}
\]
An empty supremum here is zero. In particular arbitrary central orthogonal families of corner unitaries assemble by C02 to a unitary on the sum of their central units. The uniform norm bounds used by averaging survive this exact assembly.
The supremum of any family of finite **central** projections is finite. To see this without assuming directed finite joins, well-order the family \((z_\alpha)\) by ZFC and form
\(d_\alpha=z_\alpha(1-\bigvee_{\beta<\alpha}z_\beta)\). These are pairwise orthogonal central projections, each finite because it is under \(z_\alpha\). Their sum is the original join: by transfinite induction the joins of the two families agree at every initial segment, including limit segments by C02. Apply the central patching argument to the \(d_\alpha\). This assertion concerns central projections; arbitrary infinite orthogonal sums of finite projections need not be finite.
## C10. Finite orthogonal sums in an arbitrary algebra
Let \(p,q\in M\) be finite and orthogonal. C05 gives a central \(z\) such that
\(zp\precsim zq\) and \((1-z)q\precsim(1-z)p\).
On the first component put \(a=zp\), \(b=zq\), and choose \(v\) with
\(v^*v=a\), \(vv^*=r\le b\). In \(M_2(M)\), take the concrete column operator
\[
T=\begin{pmatrix}v&0\\b&0\end{pmatrix}.
\]
Because \(a\perp b\) and \(v=va\), \(vb=0\) and \(bv^*=0\). Consequently
\[
T^*T=\begin{pmatrix}a+b&0\\0&0\end{pmatrix},\qquad
TT^*=\begin{pmatrix}r&0\\0&b\end{pmatrix}
\le\begin{pmatrix}b&0\\0&b\end{pmatrix}.
\tag{C10.1}
\]
The last projection is finite by C08, because \(b\le q\) is finite by C03. Hence C03 in \(M_2(M)\) makes \(\operatorname{diag}(a+b,0)\) finite. If \(a+b\) were not finite in \(M\), embedding an isometry of its corner as \(\operatorname{diag}(s,0)\) would contradict that matrix projection's finiteness. Thus \(z(p+q)\) is finite. The second component uses the same column with \(a=(1-z)q\), \(b=(1-z)p\), and gives finiteness of \((1-z)(p+q)\). C09 patches the two components, proving \(p+q\) finite.
Induction proves that every finite orthogonal sum of finite projections is finite. The empty sum is zero, already finite. No assertion about an arbitrary infinite orthogonal sum follows. The logical order is C05 → C06 in an already finite algebra → C07 → C08 → C10 in an arbitrary algebra. In particular C06 does not presuppose C10; there is no finite-sum/complement cycle.
## C11. Finite joins and complements of equivalent finite projections
For arbitrary \(p,q\), apply C01 to \(x=(1-p)q\). Its kernel is the orthogonal direct sum
\((1-q)H\oplus(qH\cap pH)\), so its initial support is \(q-p\wedge q\). Its range is contained in \((p\vee q-p)H\). To prove density there, a vector \(\eta\) in that subspace perpendicular to \((1-p)qH\) has \(p\eta=0\) and \(q\eta=0\), so it is perpendicular to both ranges defining \(p\vee q\), and therefore is zero. Thus
\[
p\vee q-p\sim q-p\wedge q.
\tag{C11.1}
\]
If \(p,q\) are finite, the right side is finite by C03, hence so is the left. It is orthogonal to \(p\), and C10 makes their sum \(p\vee q\) finite. Induction gives finite joins of any finite family of finite projections.
If \(p,q\) are finite and equivalent in an arbitrary \(M\), set \(e=p\vee q\). It is finite. C06 in \(eMe\) matches \(e-p\) with \(e-q\); adding the identity on \(1-e\) matches \(1-p\) with \(1-q\). Adding this match to the original \(v\) gives a unitary \(u\in M\) with \(upu^*=q\). This is the stronger arbitrary-ambient form, proved after finite joins. It is kept separate from the earlier finite-ambient C06 used in the noncircular matrix proof.
## C12. The exact semifinite projection consequence
If **semifinite** is defined to mean that each nonzero projection contains a nonzero finite projection, this property is a hypothesis. It gives a net of finite projections increasing strongly to one: the set of finite projections is directed by C11's finite join; its join must be one, since a nonzero complementary projection would contain another nonzero finite projection. C02 gives strong convergence of this directed net.
If instead semifinite is defined to mean that \(1=\sum_{i\in I}p_i\) for an orthogonal family of finite projections, the nonzero-corner property follows from C04 and C03. For nonzero \(q\), some \(qp_i\ne0\); otherwise the finite sums converge strongly to one and \(q\) would be zero. Thus \(qMp_i\ne0\), and C04 provides a nonzero subprojection of \(q\) equivalent to a subprojection of \(p_i\), hence finite. Conversely, the nonzero-corner property gives such an orthogonal family by Zorn: a maximal orthogonal family of nonzero finite projections has zero complementary projection. C10 makes each finite partial sum finite, and C02 gives their strong limit one. Both definitions therefore provide the exact SF projection input under either convention, including arbitrary cardinality and the zero algebra.
If the word instead presupposes a faithful normal semifinite **weight**, its equivalence to these projection definitions is a separate weight-theoretic prerequisite and is not proved here. No measure-theoretic finiteness of a weight is substituted for projection finiteness.
## Finite homogeneous components and monic projections
In D04–D10 the ambient algebra is finite. A projection is abelian when its corner is commutative. A nonzero projection is monic when it belongs to a finite orthogonal family of equivalent projections summing to its central support in this ambient algebra.
## D01. Finite projection approximants from continuous cutoffs
For a self-adjoint \(h\in eMe\), set
\[
E_t=s((h-te)_+)\le e\qquad(t\in\mathbb R).
\]
The continuous positive part is supplied by F06–F08 and its support by H02. Every \(E_t\) commutes with \(h\) and with every \(E_s\): the resolvent construction of a support commutes with any operator commuting with the supported positive element, and the relevant continuous functions of \(h\) commute. Also \(E_t\le E_s\) for \(s\|h\|\) and a finite grid \(-C=t_0
## D02. Abelian projections and their ambient centers
The following facts hold in an arbitrary ambient \(M\), without finiteness. If \(p\) is abelian and \(r\le p\), then
\[
c(r)p=r.
\tag{D02.1}
\]
Indeed \(rM(p-r)=r(pMp)(p-r)=0\) by commutativity. C04 gives \(c(r)c(p-r)=0\), so \(c(r)(p-r)=0\), whereas \(c(r)r=r\).
For a central projection \(z\), one also has
\[
c(zp)=zc(p).
\tag{D02.2}
\]
The right side is a central majorant of \(zp\). If a central projection \(d\) majorizes \(zp\), then \(d\vee(1-z)\) majorizes \(p\); multiplying its majorization of \(c(p)\) by \(z\) gives \(zc(p)\le d\). This proves minimality.
The map
\[
\theta:Z(M)c(p)\longrightarrow pMp,\qquad a\longmapsto ap
\tag{D02.3}
\]
is a unital *-isomorphism. It is a homomorphism because \(a\) is central. If \(ap=0\), then \(a\) annihilates every vector \(xp\xi\), since \(axp\xi=xap\xi=0\); C04's dense range description implies \(ac(p)=0\), hence \(a=0\). Thus F05–F07 make \(\theta\) isometric with closed range. Every projection \(r\le p\) is in that range by (D02.1). D01's finite projection approximants make its range all of \(pMp\).
Abelian projections are finite: an isometry in the commutative corner has \(vv^*=v^*v=p\). Abelianity is hereditary and preserved by equivalence, by C03's corner maps. A sum of abelian projections with mutually orthogonal central supports is abelian. To check the latter, put \(p=\sum_i p_i\) and \(z_i=c(p_i)\). On each \(z_i\), elements of \(pMp\) lie in \(p_iMp_i\) and commute. The strong sum of the \(z_i\) supports \(p\), so taking finite central sums shows that their full commutator is zero.
Two abelian projections with the same central support are equivalent. Apply C05 to \(p,q\) with \(c(p)=c(q)=c\). On a central comparison piece \(z\le c\), suppose \(pz\sim r\le qz\). Then \(c(r)=c(pz)=z\) by equivalence and (D02.2), so (D02.1) for \(q\) gives \(r=c(r)q=qz\). Thus the subequivalence is equivalence on this piece; the reversed piece is treated the same way, and C02 adds them. This also handles zero comparison pieces.
## D03. Every type I corner has an abelian projection of full support
Here **type I** means every nonzero projection has a nonzero abelian subprojection. If \(r\ne0\) lies in a type I algebra, choose by Zorn a maximal family of nonzero abelian \(q_i\le r\) whose central supports are mutually orthogonal. Its sum \(q\) is abelian by D02, and
\(c(q)=\bigvee_i c(q_i)\): any central majorant of the sum majorizes each summand and conversely. This equals \(c(r)\). Otherwise the nonzero central projection
\(z=c(r)-\bigvee_i c(q_i)\) has \(rz\ne0\), by minimality of \(c(r)\). Type I supplies a nonzero abelian projection under \(rz\), with central support at most \(z\), contradicting maximality. Thus \(q\le r\) is abelian with \(c(q)=c(r)\). The family need not be countable.
## D04. No infinite orthogonal family of equivalent nonzero projections in a finite algebra
If an orthogonal family \((p_i)_{i\in I}\) of nonzero projections in a finite \(M\) is infinite and all its members are equivalent, ZFC selects distinct members \(p_0,p_1,\ldots\). Choose partial isometries matching \(p_n\) onto \(p_{n+1}\). C02's sum has initial projection \(e=\sum_{n\ge0}p_n\) and final projection \(e-p_0
## D05. Finite type I components are centrally homogeneous
Let \(M\) be finite and type I, and choose a nonzero abelian \(p\). A maximal orthogonal family \(p_1,\ldots,p_n\) of projections equivalent to \(p\), containing \(p\), exists by Zorn and is finite by D04. Its members are abelian by D02, each has central support \(c=c(p)\), and their sum \(e\) is at most \(c\).
If the residual \(r=c-e\) had \(c(r)=c\), D03 would give an abelian \(q\le r\) with \(c(q)=c\). D02 would give \(q\sim p\), contradicting maximality. Hence \(c(r)0,\qquad \sum_{i=1}^n zp_i=z.
\tag{D05.1}
\]
The projections \(zp_i\) are abelian, equivalent, and nonzero, since each has central support \(z\) by (D02.2). Thus \(Mz\) is a finite homogeneous component with \(n\) equivalent abelian projections summing to its unit.
Now take a maximal orthogonal family of nonzero central homogeneous components of a finite type I algebra. Its central sum is one: if the complement were nonzero, it would remain finite and type I and the preceding construction would supply another component. This again uses Zorn on a fixed set of projections and their finite implementing families. For each positive integer \(n\), group the components having \(n\) members. Their sum \(z_n\) is central. On that sum, add the first, second, ..., \(n\)-th members of their respective families separately. C02's sums of the matching partial isometries give \(n\) equivalent projections, D02's central sums make them abelian, and their sum is \(z_n\). Consequently
\[
1=\sum_{n\ge1}z_n,
\qquad Mz_n\text{ has }n\text{ equivalent abelian projections summing to }z_n
\tag{D05.2}
\]
for each nonzero \(z_n\). The original collection of components can be uncountable; grouping by \(n\) does not assert a countable central-support family. Uniqueness of the multiplicity labels is not needed or asserted by this existence construction.
## D06. Exact matrix-unit realization on a homogeneous piece
Suppose \(p_1,\ldots,p_n\) are equivalent abelian projections summing to a central \(z\). Their common central support is \(z\), because the sum is \(z\). Choose \(v_i\) with \(v_i^*v_i=p_1\), \(v_iv_i^*=p_i\), and \(v_1=p_1\). Then
\(e_{ij}=v_iv_j^*\) satisfy \(e_{ij}^*=e_{ji}\), \(e_{ij}e_{kl}=\delta_{jk}e_{il}\), and \(\sum_i e_{ii}=z\). The maps
\[
Mz\longrightarrow M_n(p_1Mp_1),\quad x\longmapsto[v_i^*xv_j]_{ij},
\qquad
[a_{ij}]\longmapsto\sum_{i,j=1}^n v_i a_{ij}v_j^*
\tag{D06.1}
\]
are mutually inverse unital *-homomorphisms. To verify multiplicativity, insert \(\sum_k v_kv_k^*=z\) between the factors; to verify the inverse, use \(v_j^*v_k=\delta_{jk}p_1\) and \(zxz=x\). All sums are finite, so there is no missing convergence or surjectivity assertion. D02 identifies \(p_1Mp_1\) isometrically with \(Z(M)z\); hence
\[
Mz\cong M_n(Z(M)z).
\tag{D06.2}
\]
F05–F07 give exact norm preservation of these algebraic *-isomorphisms. This is the finite homogeneous matrix identification consumed by the averaging argument. It uses no character theorem, direct-integral classification or representation-independence theorem.
The center of \(Mz\) is exactly \(Z(M)z\), since a central corner element extended by zero commutes with both central pieces of \(M\). At the concrete level the maps in (D06.1) are finite sums of bounded left/right multiplication and therefore are ultraweak continuous by H03's series-vector formulas. The center/corner identification (D02.3) has the following direct bounded weak-operator inverse estimate: approximate vectors of \(zH\) by finite sums \(xp_1\xi\), using C04. For such vectors,
\[
\langle xp_1\xi,a yp_1\eta\rangle
=\langle p_1\xi,(ap_1)(p_1x^*yp_1\eta)\rangle
\qquad(a\in Z(M)z).
\]
Thus bounded weak-operator convergence of \(ap_1\) implies that of \(a\), first on a dense family and then everywhere. H03 turns bounded weak-operator convergence into ultraweak convergence.
Full ultraweak continuity of this specific inverse follows directly as well. H03 constructs a Banach predual \(Q\) for each concrete corner; its canonical image in the full norm dual is isometric by F01's norming-functional theorem, and hence norm closed. Therefore the corner's series-vector functionals are norm closed. Given a vector coefficient \(a\mapsto\langle\xi,a\eta\rangle\) on \(Z(M)z\), replace \(\xi,\eta\in zH\) by finite sums of vectors \(xp_1\alpha\). The preceding coefficient identity expresses each resulting approximant, after applying \(\theta^{-1}\), as a finite sum of vector coefficients on \(p_1Mp_1\). Vector-norm approximation makes the original functionals converge in functional norm. Because \(\theta^{-1}\) is isometric, their transported functionals converge in the same norm, so norm closure makes the transported coefficient normal. For a series-vector functional, first truncate the series, with functional-norm tail bounded by the sum of the products of the vector norms, and then use the same argument. Hence every concrete normal functional composed with \(\theta^{-1}\) is normal, proving ultraweak continuity for this inverse on arbitrary nets. Forward continuity is bounded multiplication by \(p_1\), directly in the series-vector formula. This is a local proved normality statement for (D02.3), rather than a general automatic-normality import. C01's finite-vector argument justifies applying H03 to all the concrete corners and centers involved. Algebraic order and all existing positive suprema are also preserved by the *-isomorphism and its inverse by F08 and the definition of a supremum.
## D07. The finite algebra splits into these pieces and a no-abelian piece
In an arbitrary \(M\), let \(z_I\) be the join of all abelian projections. The family is preserved by every unitary conjugation, so C04's unitary-span argument makes \(z_I\) central. If \(0\ne q\le z_I\), then \(qr\ne0\) for some abelian \(r\): otherwise \(q\) would vanish on their joined range and satisfy \(qz_I=0\). C04's polar contact gives a nonzero subprojection of \(q\) equivalent to a subprojection of \(r\), hence abelian by D02. Therefore \(Mz_I\) is type I. Its complement has no nonzero abelian projection by definition.
For finite \(M\), both central pieces are finite by C03. Apply D05–D06 to \(Mz_I\). We obtain arbitrary central homogeneous matrix pieces, grouped as in (D05.2), together with \(M(1-z_I)\), which is finite and has no nonzero abelian projection. This is the finite central decomposition required here. No infinite or general type classification follows from it.
## D08. Halving and coherent dyadic partitions on the no-abelian piece
Suppose \(N\) has no nonzero abelian projection. Every nonzero corner \(rNr\) is then nonabelian. D01 implies that some projection \(a\le r\) is not central in that corner. Hence \(aN(r-a)\ne0\): if this corner and its adjoint were zero, then \(a\) would commute with every element of \(rNr\). C04 gives nonzero equivalent subprojections under \(a\) and \(r-a\), which are orthogonal.
For a given projection \(r\), choose by Zorn a maximal family of these equivalent pairs \((a_i,b_i)\) with all the projections in the combined family mutually orthogonal and under \(r\). C02 gives \(A=\sum_i a_i\sim B=\sum_i b_i\). A nonzero residual \(r-A-B\) would yield another pair by the preceding paragraph, so
\[
r=A+B,\qquad A\sim B.
\tag{D08.1}
\]
The zero corner has the zero pair. For a nonzero corner both halves are nonzero.
In a finite \(N\) without nonzero abelian projections, (D08.1) can be iterated coherently. At level \(m\), choose an orthogonal partition \((e_{m,k})_{0\le k<2^m}\) into equivalent projections and matching \(v_{m,k}\) from \(e_{m,0}\) onto \(e_{m,k}\), with the first matching the identity. Halve \(e_{m,0}=f_0+f_1\) by (D08.1), and define
\[
e_{m+1,2k+j}=v_{m,k}f_jv_{m,k}^*\qquad(j=0,1).
\]
These projections remain pairwise orthogonal and equivalent, sum to the unit, and refine the old partition. With
\(P_{k/2^m}=\sum_{i
## D09. A dyadic central cut fits under every nonzero projection
Let finite \(N\) have no nonzero abelian projection, and let \(p\ne0\). Work on the central support \(c(p)\), and restrict the partitions in D08 to this central unit. There is a nonzero central \(z\le c(p)\) and some \(m\ge0\) such that
\[
q_m z\precsim pz.
\tag{D09.1}
\]
If not, apply C05 to \(q_m c(p)\) and \(p\) for each \(m\). Its first comparison part must be zero: on every nonzero central part \(z\le c(p)\), both \(q_mz\) and \(pz\) are nonzero by their full central supports, so that part would give (D09.1). Consequently \(p\precsim q_m c(p)\) for every \(m\). By (D08.2),
\(p\precsim (q_m-q_{m+1})c(p)\) for every \(m\ge0\). Those annular projections are mutually orthogonal. Choosing a copy of \(p\) under each yields infinitely many mutually orthogonal nonzero equivalent projections, contradicting D04. This proves (D09.1) without a trace-size estimate.
A nonzero subprojection of \(pz\) equivalent to \(q_mz\) is monic: the central cut of the dyadic partition supplies a finite family of equivalent copies summing to \(z\). Equivalence preserves this property. If a partition containing the chosen copy itself is required, C06 extends its equivalence with \(q_mz\) to a unitary, which conjugates the partition and fixes its central sum.
## D10. Monic subprojections and arbitrary monic decompositions
On a homogeneous piece with unit \(z\) and abelian cells \(e_1,\ldots,e_n\), every nonzero \(p\le z\) has a nonzero monic subprojection. Since \(c(e_1)=z\), C04 gives a nonzero partial-isometry match from a subprojection \(s\le e_1\) to a subprojection \(r\le p\). Let \(w=c(s)>0\). D02 gives \(s=we_1\), and the central restrictions \(we_i\) are \(n\) equivalent cells summing to \(w\). Thus \(r\sim s\) is monic in the original ambient algebra. C06 extends this equivalence to a unitary of the finite algebra; conjugating the finite partition then makes \(r\) itself one of its cells while retaining the central sum.
For a general finite \(M\), use D07. A nonzero \(p\) has a nonzero central restriction either on a homogeneous piece or on the piece without nonzero abelian projections. The first case uses the preceding paragraph, and the second uses D09. All central units of these components and their further central cuts are central in the original \(M\). Hence every nonzero \(p\) contains a nonzero monic projection relative to \(Z(M)\).
Choose by Zorn a maximal orthogonal family \((r_i)\) of nonzero monic subprojections of \(p\). Its strong sum is \(p\): a nonzero residual would contain another such projection. Empty families cover \(p=0\). Thus every projection in a finite \(M\) is an arbitrary orthogonal sum of monic projections. No countability of this decomposition is asserted.
## The finite spectral cuts used by the constructions
Continuous positive parts and their closed-range supports give every finite cut below. The endpoint conventions are recorded explicitly, so the constructions apply to eigenvalues at the cut endpoints as well.
## S01. Strict and non-strict threshold projections
For \(h=h^*\in M\) and \(t\in\mathbb R\), set
\[
a_t=(h-t1)_+,\qquad b_t=(t1-h)_+,\qquad
P_t=s(a_t),\qquad Q_t=1-s(b_t),
\]
where \(s(a)\) is the projection onto \(\overline{\operatorname{ran}a}\) supplied by [H02](regular-group-operator-foundations.md#h02). Continuous positive parts give
\[
h-t1=a_t-b_t,\qquad a_tb_t=b_ta_t=0.
\]
All these operators belong to \(M\). Indeed, continuous functions of \(h\) are norm limits of polynomials. For \(c\ge0\), H02 gives the exact formula
\[
s(c)=1-\underset{r\to\infty}{\operatorname{s-lim}}(1+rc)^{-1}.
\]
Each inverse belongs to \(M\) by H01. Apply this formula to \(a_t\) and \(b_t\). The algebra is closed under both norm and strong limits.
An operator commuting with \(h\) commutes with \(a_t,b_t\), their resolvents, and their support projections. This follows first for polynomials, then for their norm limits, and finally for the strong limits, since multiplication by a fixed bounded operator preserves strong convergence. Consequently, all the projections \(P_t,Q_t\) commute with \(h\), with each other, and with every operator commuting with \(h\). If \(h\) is central in \(M\), the projections are central in \(M\).
The closed ranges of \(a_t\) and \(b_t\) are orthogonal: for every \(\xi,\eta\in H\),
\[
\langle a_t\xi,b_t\eta\rangle
=\langle\xi,a_tb_t\eta\rangle=0.
\]
Thus \(s(a_t)s(b_t)=0\), or \(P_t\le Q_t\). Compression by a commuting projection preserves positivity: if \(c\ge0\) commutes with a projection \(r\), then \(cr=c^{1/2}rc^{1/2}\ge0\). It follows that
\[
(h-t1)P_t\ge0,\qquad (h-t1)(1-P_t)\le0,
\]
\[
(h-t1)Q_t\ge0,\qquad (h-t1)(1-Q_t)\le0.
\]
For example, \((h-t1)P_t=a_t\), because \(a_tP_t=a_t\) and \(b_tP_t=0\); on \(1-P_t\), only the non-positive term remains. The corresponding statements for \(Q_t\) follow because \(b_tQ_t=0\).
The projection
\[
E_t=Q_t-P_t=1-s(a_t)-s(b_t)
\]
is exactly the projection onto \(\ker(h-t1)\). One inclusion follows from \(a_tE_t=b_tE_t=0\). For the other, \((h-t1)\xi=0\) gives \(a_t\xi=b_t\xi\); these two vectors lie in orthogonal ranges, so both vanish. Thus \(P_t\) excludes the eigenspace at \(t\), whereas \(Q_t\) includes it. These are, respectively, the strict cut above \(t\) and the non-strict cut at \(t\).
## S02. Nesting and exact interval endpoints
If \(0\le a\le b\), then \(\ker b\subseteq\ker a\). In fact, \(b\xi=0\) gives
\[
0\le\langle a\xi,\xi\rangle\le\langle b\xi,\xi\rangle=0,
\]
and the positive-form argument in H02 gives \(a\xi=0\). Taking orthogonal complements proves \(s(a)\le s(b)\).
For \(s0\) and \(r\ge0\), this implies \(r=0\), as required.
For \(s
## S03. Uniform approximation by finitely many projections
Every bounded selfadjoint \(h\in M\) is a norm limit of finite real linear combinations of mutually orthogonal projections in \(M\) commuting with \(h\). If \(h=0\), the single combination \(0\cdot1\) suffices. Otherwise put \(R=\|h\|>0\). For a positive integer \(m\), let
\[
t_j=-R+\frac{2Rj}{m}\quad(0\le j\le m),
\qquad \Delta=\frac{2R}{m}.
\]
The norm and order conclusions of F06–F08 give \(-R1\le h\le R1\), so \(Q_{-R}=1\), \(P_R=0\), and \(Q_R=E_R\). Define
\[
p_j=Q_{t_j}-Q_{t_{j+1}}\quad(0\le j2\|h\|/\varepsilon\) gives an approximation with error strictly less than \(\varepsilon\). Zero pieces may be omitted. The argument works in arbitrary Hilbert-space dimension and introduces no countable spectral resolution.
For a positive element the approximants can themselves be chosen positive. If \(h\ge0\) and \(R=\|h\|>0\), instead use \(t_j=Rj/m\), \(0\le j\le m\). Now \(Q_0=1\) and \(Q_R=E_R\); the same differences \(p_j=Q_{t_j}-Q_{t_{j+1}}\) and top projection \(p_m=Q_R\) sum to one. Their coefficients \(t_j\) and \(R\) are nonnegative, and the same interval bounds give
\[
0\le h_m=\sum_{jStrict threshold endpoints and the finite lower-step approximation
The upper panel shows the endpoint conventions proved in S01–S02. The lower panel shows the scalar example \(R=1,m=4\) of S03: each half-open interval receives its lower endpoint coefficient, and \(\{1\}\) receives coefficient one. The shaded difference is at most \(\Delta=1/2\). These scalar diagrams explain the operator construction from [continuous positive parts and [range supports](regular-group-operator-foundations.md#h02); the displayed norm estimate is proved for every bounded selfadjoint operator.
## S04. Central cuts for a positive contraction
Let \(a\in Z(M)\) satisfy \(0\le a\le1\), and let \(N\ge1\) be an integer. Apply S01 to \(a\), writing its projections as \(Q_t\). Then
\[
z_k=Q_{k/N}-Q_{(k+1)/N}\quad(0\le k
## S05. Integer central ranks without a representation theorem
Let \(D\) be a nonzero unital abelian \(C^*\)-algebra, let \(n\ge1\), and let \(p=(p_{ij})\) be a projection in \(M_n(D)\). Its ordinary diagonal trace is
\[
d=\sum_{i=1}^n p_{ii}\in D.
\]
It is selfadjoint. The full character construction in [T05a and CS01–CS06](regular-group-operator-foundations.md#t05a) supplies characters separating the elements of \(D\), and every character \(\chi\) extends entrywise to a unital \(*\)-homomorphism \(\chi_n:M_n(D)\to M_n(\mathbb C)\).
For each \(\chi\), the numerical matrix \(\chi_n(p)\) is a selfadjoint projection. Its trace is an integer between zero and \(n\). Here is the finite-dimensional argument. Apply Gram–Schmidt successively to its images of the standard basis vectors, discarding zero remainders, to obtain an orthonormal basis for its range. Do the same with the images under its complementary projection to obtain an orthonormal basis for its kernel. These two subspaces are orthogonal and together span \(\mathbb C^n\). In the combined basis the projection has \(r\) diagonal entries equal to one and the others zero. Its trace is unchanged by this basis change, since finite sums show \(\operatorname{Tr}(AB)=\operatorname{Tr}(BA)\), and hence \(\operatorname{Tr}(U^*PU)=\operatorname{Tr}(PUU^*)\). Thus
\[
\chi(d)=\operatorname{Tr}(\chi_n(p))\in\{0,1,\ldots,n\}.
\]
Put \(q(X)=\prod_{j=0}^n(X-j)\). Every character vanishes on \(q(d)\), so character separation gives \(q(d)=0\). This proves the spectrum assertion directly. If \(\lambda\notin\{0,\ldots,n\}\), the polynomial identity
\[
q(X)-q(\lambda)=(X-\lambda)r_\lambda(X)
\]
gives
\[
(d-\lambda1)\left(-\frac{r_\lambda(d)}{q(\lambda)}\right)
=\left(-\frac{r_\lambda(d)}{q(\lambda)}\right)(d-\lambda1)=1.
\]
Consequently \(\sigma_D(d)\subseteq\{0,1,\ldots,n\}\).
There is also an explicit decomposition into central rank pieces. For \(0\le k\le n\), let
\[
L_k(X)=\prod_{\substack{0\le j\le n\\j\ne k}}
\frac{X-j}{k-j},\qquad w_k=L_k(d).
\]
These have real coefficients, so the \(w_k\) are selfadjoint. For each character,
\[
\chi(w_k)=
\begin{cases}1,&\chi(d)=k,\\0,&\chi(d)\ne k.\end{cases}
\]
Applying character separation to each algebraic identity gives
\[
w_k^2=w_k,\quad w_kw_l=0\ (k\ne l),\quad
\sum_{k=0}^n w_k=1,\quad (d-k1)w_k=0.
\]
Therefore
\[
d=\sum_{k=0}^n k w_k,
\qquad 0\le d\le n1.
\]
Some rank pieces may be zero. When \(D=Z(M)\), these are central projections in \(M\). If the already established matrix-corner trace formula is \(T(p)=d/n\), they satisfy \(T(p)w_k=(k/n)w_k\), exactly as needed by finite matrix averaging. This deduction uses that formula as a separate hypothesis; it does not establish the centre-valued trace or the alignment of equivalent projections. No normality of the characters is needed. If \(D\) is the zero algebra, \(p=d=w_k=0\) and the assertion is immediate, without characters.
## S06. Projections determine bounded linear functionals
Let \(f,g:M\to\mathbb C\) be bounded linear functionals agreeing on every projection. For \(h=h^*\), S03 gives finite projection combinations \(h_m\) with \(\|h-h_m\|\to0\). Linearity gives \(f(h_m)=g(h_m)\), and
\[
|f(h)-g(h)|\le(\|f\|+\|g\|)\|h-h_m\|\longrightarrow0.
\]
Every \(x\in M\) is \((x+x^*)/2+i(x-x^*)/(2i)\), a sum of two selfadjoint parts, so \(f=g\) on \(M\).
For completeness, a positive complex-linear functional \(\varphi\) with \(\varphi(1)=1\) is automatically bounded with norm one. Positive parts give \(\varphi(h)\in\mathbb R\) for selfadjoint \(h\), and then \(\varphi(x^*)=\overline{\varphi(x)}\). The sesquilinear form \((a,b)\mapsto\varphi(b^*a)\) is positive. Put \(\alpha=\varphi(b^*a)\), \(\beta=\varphi(b^*b)\), and \(\gamma=\varphi(a^*a)\). For every complex \(t\), positivity gives
\[
0\le\gamma-\overline t\alpha-t\overline\alpha+|t|^2\beta.
\]
If \(\beta>0\), take \(t=\alpha/\beta\). If \(\beta=0\), take \(t=r\alpha\) with \(r>0\); then \(0\le\gamma-2r|\alpha|^2\) for every \(r\), forcing \(\alpha=0\). These two cases prove
\[
|\varphi(b^*a)|^2\le\varphi(a^*a)\varphi(b^*b).
\]
Set \(b=1\). Since \(0\le a^*a\le\|a\|^2 1\),
\[
|\varphi(a)|^2\le\varphi(a^*a)\le\|a\|^2.
\]
Thus \(\|\varphi\|\le1\), and evaluation at \(1\) gives equality.
In particular, if \(\rho\) is a tracial state and \(T:M\to Z(M)\) is positive and unital, then \(\rho\circ T\) is also a positive unital scalar functional, hence has norm one. Agreement of \(\rho\) and \(\rho\circ T\) on projections implies agreement everywhere by S03 and the estimate
\[
|\rho(h)-\rho(T(h))|\le2\|h-h_m\|.
\]
This estimate needs neither normality of \(\rho\) nor a separately imported norm estimate for \(T\).
## Constructing the normal center-valued trace
The proofs in this section use C05–C06, D10 and the support-based finite spectral approximation S01–S03. They do not presuppose a scalar trace on a general finite algebra.
**Norm of a positive central map (CN).** If a positive linear map \(F:A\to Z\) has commutative unital C*-algebra range and \(F(1)=c\ge0\), then \(\|F\|=\|c\|\).
**Proof.** For every character \(\chi\) of \(Z\), \(\chi F\) is a positive scalar functional. Its positive-form Cauchy–Schwarz inequality gives
\[
|\chi F(x)|^2\le \chi(c)\,\chi F(x^*x)
\le \chi(c)^2\|x\|^2.
\]
The inequality \(x^*x\le\|x\|^2 1\) is F08. Characters preserve positivity by CS04 and norm \(Z\) by CS05 of [Regular-group operator foundations](regular-group-operator-foundations.md#t05a). Hence \(\|F(x)\|\le\|c\|\|x\|\); evaluation at the unit proves equality. The positive-form Cauchy–Schwarz argument, including its zero-denominator case, is the quotient argument in [GNS](hypertraces-finite-injectivity.md#state-quotient). \(\square\)
**Assembly on central components (CA).** Suppose \((z_\alpha)\) is an arbitrary orthogonal central partition of the unit, and each \(F_\alpha:Mz_\alpha\to Zz_\alpha\) is normal and has norm at most \(C\). The componentwise map \(F:M\to Z\) is bounded by \(C\) and normal.
**Proof.** The strong orthogonal sum of the output components exists, with norm the supremum of their norms, by H00–H01 and the central product construction SUP in [Hypertraces](hypertraces-finite-injectivity.md#normal-central-supports). It is linear and bounded. A series-vector normal test on \(Z\) is
\(\eta(d)=\sum_j\langle\xi_j,d\eta_j\rangle\), with \(\sum_j\|\xi_j\|\|\eta_j\|<\infty\). Its \(\alpha\)-component has norm at most \(\sum_j\|z_\alpha\xi_j\|\|z_\alpha\eta_j\|\). Thus
\[
\sum_\alpha\|\eta|_{Zz_\alpha}\|
\le\sum_{j,\alpha}\|z_\alpha\xi_j\|\|z_\alpha\eta_j\|
\le\sum_j\|\xi_j\|\|\eta_j\|.
\]
The last inequality is Cauchy–Schwarz on the orthogonal coordinate families. These sums have countably many nonzero terms, even for an uncountable partition. Pulling each component test back by its normal map and normal central compression gives normal functionals on \(M\) with summable norms. Their sum belongs to the norm-closed concrete predual H03, and equals \(\eta F\). All normal tests on \(Z\) therefore pull back to normal tests on \(M\), which proves ultraweak continuity. \(\square\)
*Figure 2. The central state, residual matching, finite comparison constant, monic copying and norm limit in NC–CT. The strict central inequalities are interpreted on their stated supports in LC. The argument follows the approximate-trace method in Peterson, Lemmas 6.4.8–6.4.9 and Theorem 6.4.10, with a separate construction of the finite constant in LC1.*
**A normal center-valued state (NC).** Every concrete von Neumann algebra \(M\), with center \(Z\), has a normal positive unital center-linear map \(\Phi:M\to Z\).
**Proof.** SUP constructs an orthogonal central partition \((z_\alpha)\) from the supports of normal vector states of \(Z\). On each \(Zz_\alpha\) the corresponding vector state is faithful. To treat one component, write its unit as \(z\), choose the normalized vector \(\xi=z\xi\), and put \(K=\overline{Zz\xi}\). Its vector state \(\omega(d)=\langle\xi,d\xi\rangle\) on \(Zz\) is faithful, normal and tracial, because this algebra is abelian. The map \(d1\mapsto d\xi\) identifies its trace Hilbert completion with \(K\). The full normal realization proof REP in [Hypertraces](hypertraces-finite-injectivity.md#trace-realization) applies to this faithful normal trace. In an abelian algebra its left and right multiplications coincide, so REP proves
\[
\theta(Zz)'=\theta(Zz),\qquad \theta(d)=d|_K,
\]
and proves that \(\theta\) and its inverse are normal. This invocation of REP uses only its preceding scalar, Hilbert-space and predual constructions, and no center-valued trace theorem.
Let \(P\) be the projection onto \(K\). The subspace reduces \(Z\), so \(P\) commutes with it. If \(x\in Mz\), the compression \(PxP|_K\) commutes with \(\theta(Zz)\), and hence belongs to \(\theta(Zz)\). Define
\[
\Phi_z(x)=\theta^{-1}(PxP|_K).
\]
Compression is positive, unital and normal by H03's series-vector tests. The inverse \(\theta^{-1}\) is a normal *-isomorphism, so this map is positive, unital and normal. Commutation with \(Z\) proves \(\Phi_z(dx)=d\Phi_z(x)\). CN gives its norm one. Assemble these maps by CA. Their componentwise positivity, unit value and center-linearity survive assembly. The zero algebra uses its zero map. \(\square\)
**An almost tracial corner (LC).** Let \(M\ne0\) be finite, let \(\Phi:M\to Z\) be a normal center-valued state, and let \(\varepsilon>0\). There is a nonzero projection \(p\) such that \(\Phi\) is faithful on \(pMp\) and
\[
\Phi(xx^*)\le(1+\varepsilon)\Phi(x^*x)\qquad(x\in pMp).
\tag{LC}
\]
**Proof.** Choose a maximal orthogonal family of projections \(q_i\) with \(\Phi(q_i)=0\), and set \(q_0=1-\sum_i q_i\). Normality gives \(\Phi(q_0)=1\). The restriction of \(\Phi\) to \(q_0Mq_0\) is faithful: a nonzero positive element of weight zero has a nonzero positive spectral threshold projection of weight zero in that corner, contradicting maximality. The threshold projection and its lower order bound are supplied by S01–S03. Center-linearity also shows that, for a nonzero projection \(a\le q_0\), the central support of the positive element \(\Phi(a)\) is \(c(a)\). Indeed a central cut of \(\Phi(a)\) vanishes exactly when that cut of \(a\) vanishes, by faithfulness.
We first obtain a finite comparison constant. Fix a real \(C>1\). Choose a maximal family of pairs \((e_i,f_i)\) of equivalent nonzero subprojections of \(q_0\), with each family separately orthogonal, such that
\(\Phi(e_i)-C\Phi(f_i)\) is positive with support their common central support. Such families are ordered by inclusion; unions of chains give upper bounds, so Zorn applies. Put
\[
e_0=q_0-\sum_i e_i,\qquad f_0=q_0-\sum_i f_i.
\]
Orthogonal additivity of equivalence and finite-complement matching give \(e_0\sim f_0\). Also \(f_0\ne0\): otherwise normality would give
\(1\ge\Phi(\sum_i e_i)\ge C\Phi(\sum_i f_i)=C1\), which is impossible. Hence \(e_0\ne0\).
For every \(a\le e_0\), \(b\le f_0\) with \(a\sim b\),
\[
\Phi(a)\le C\Phi(b).
\tag{LC1}
\]
If this failed, the support of the positive part of \(\Phi(a)-C\Phi(b)\) would give a nonzero central cut on which the inequality is strictly reversed. The equivalent cut projections would extend the maximal family. Their common support is exactly this cut, since it lies under \(c(a)=c(b)\). This proves (LC1).
Let \(\mu\) be the infimum of all nonnegative real constants satisfying (LC1) on these residual corners. It is finite, at most \(C\), and its defining inequalities hold at the infimum because the positive cone is norm closed. It is positive: the full equivalent pair \(e_0,f_0\) gives \(\|\Phi(e_0)\|\le\mu\|\Phi(f_0)\|\), and both weights are nonzero. Since \(\mu/(1+\varepsilon)<\mu\), there are equivalent \(e\le e_0\), \(f\le f_0\) for which
\((1+\varepsilon)\Phi(e)\not\le\mu\Phi(f)\). Cut by the support \(z\ne0\) of the positive part of \((1+\varepsilon)\Phi(e)-\mu\Phi(f)\), replacing \(e,f\) by \(ez,fz\). On this central corner
\[
(1+\varepsilon)\Phi(e)-\mu\Phi(f)>0
\tag{LC2}
\]
means a positive element with support \(z=c(e)=c(f)\). All the upper inequalities \(\Phi(a)\le\mu\Phi(b)\) for equivalent subprojections remain valid after this cut.
Now choose a maximal separately orthogonal family \((\widehat e_i,\widehat f_i)\) of equivalent nonzero subprojections of \(e,f\), respectively, satisfying
\((1+\varepsilon)\Phi(\widehat e_i)\le\mu\Phi(\widehat f_i)\) on their common central support. Set
\[
p=e-\sum_i\widehat e_i,\qquad q=f-\sum_i\widehat f_i.
\]
They are equivalent by finite-complement matching: transport one sum by an equivalence \(e\sim f\), then match its complement in the finite corner. If \(p=0\), summing the inequalities would give \((1+\varepsilon)\Phi(e)\le\mu\Phi(f)\), contradicting (LC2). Thus \(p\sim q\ne0\). For equivalent \(a\le p\), \(b\le q\), the first residual bound and the second maximality give
\[
\Phi(a)\le\mu\Phi(b)\le(1+\varepsilon)\Phi(a).
\tag{LC3}
\]
For the second inequality, a nonzero support of the positive part of \(\mu\Phi(b)-(1+\varepsilon)\Phi(a)\) would provide another admissible central-cut pair. This is again forbidden by maximality.
If \(p_1,p_2\le p\) are equivalent, transport \(p_1\) through an equivalence \(p\sim q\) to obtain \(r\le q\) equivalent to both. Applying (LC3) to \((p_1,r)\) and \((p_2,r)\) gives
\[
\Phi(p_1)\le\mu\Phi(r)\le(1+\varepsilon)\Phi(p_2).
\]
For a unitary \(u\in pMp\), positive finite spectral sums consequently satisfy \(\Phi(udu^*)\le(1+\varepsilon)\Phi(d)\). S03's positive spectral sums converge in norm to every \(d\ge0\); boundedness CN and closedness of the positive cone preserve the inequality. Finally the polar partial isometry of \(x\in pMp\) extends to a unitary of this finite corner by finite-complement matching. Thus \(xx^*=u(x^*x)u^*\), giving (LC). Faithfulness holds because \(p\le q_0\). \(\square\)
**A normal almost trace on the whole finite algebra (NA).** For every \(a>1\), a finite von Neumann algebra has a normal center-valued state \(\Psi\) satisfying \(\Psi(xx^*)\le a\Psi(x^*x)\) for all \(x\).
**Proof.** Apply NC and LC with \(\varepsilon=a-1\). The monic-decomposition lemma supplies a nonzero monic \(h\le p\), so (LC) holds on \(hMh\). Choose equivalent orthogonal copies \(h_1,\ldots,h_k\) summing to their central unit \(z_0=c(h)\), and partial isometries \(w_i\) with \(w_i^*w_i=h_i\), \(w_iw_i^*=h\). On \(Mz_0\), define
\[
F(x)=\sum_{i=1}^k\Phi(w_ixw_i^*).
\]
This is positive, normal and center-linear. Its unit value is \(k\Phi(h)\), whose support is \(z_0\). For \(x\in Mz_0\), put \(b_{ij}=w_ixw_j^*\in hMh\). Since \(\sum_jh_j=z_0\),
\[
F(xx^*)=\sum_{i,j}\Phi(b_{ij}b_{ij}^*)
\le a\sum_{i,j}\Phi(b_{ij}^*b_{ij})=aF(x^*x).
\tag{NA1}
\]
Choose \(\delta>0\) with nonzero central threshold \(z=1_{[\delta,\infty)}(F(z_0))\). S01 gives \(F(z_0)z\ge\delta z\), so its inverse \(y\in Zz\) exists by the continuous calculus. The map \(x\mapsto yF(x)\), \(x\in Mz\), is a normal center-valued state on that component, and retains (NA1). Normality of multiplication by \(y\) follows directly from H03. Positivity follows because the two central factors commute. Center-linearity and \(yF(z)=z\) prove the unit and central identity conditions.
This construction works in every nonzero remaining central corner of \(M\). Zorn therefore gives a maximal orthogonal family of central corners with such maps, and their sum is one: a nonzero remainder would provide another corner. CA assembles them to the required normal center-valued state, whose norm is one by CN. The zero algebra is immediate. \(\square\)
**The normal center-valued trace (CT).** A finite von Neumann algebra has a unique normal positive unital center-linear tracial map \(T:M\to Z(M)\). It has norm one when the algebra is nonzero and is faithful.
**Proof.** Choose real numbers \(a_n>1\) decreasing to one and normal almost traces \(\Psi_n\) supplied by NA. For a monic projection \(h\), take equivalent copies \(h_i\), \(1\le i\le k\), summing to the central support \(z\). The almost-trace inequality applied to their implementing partial isometries gives both \(\Psi_n(h)\le a_n\Psi_n(h_i)\) and \(\Psi_n(h_i)\le a_n\Psi_n(h)\). For \(m
## S07. The two trace-factorization applications
For the normal-trace application, suppose that the centre-valued trace and the orthogonal decomposition of projections into monic projections have already been supplied. If \(p\) is monic, let \(p_1,\ldots,p_n\) be equivalent orthogonal copies of \(p\) summing to a central projection \(z\). A partial isometry \(v_i\) witnessing the equivalence has \(v_i^*v_i=p\) and \(v_iv_i^*=p_i\). Traciality gives \(\tau(p_i)=\tau(p)\) and \(T(p_i)=T(p)\). Their sums are \(\tau(z)\) and \(T(z)=z\), the latter because \(T\) is centre-linear and unital. Consequently
\[
\tau(p)=\frac{\tau(z)}n,\qquad T(p)=\frac zn.
\]
Thus \(\tau(p)=\tau(T(p))\). For an orthogonal monic decomposition \(e=\sum_{\alpha}p_\alpha\), the net \(e_G=\sum_{\alpha\in G}p_\alpha\), over finite subsets \(G\), increases strongly to \(e\). The normality clauses give \(\tau(e_G)\to\tau(e)\) and \(\tau(T(e_G))\to\tau(T(e))\); the latter uses normality of \(T\) and of \(\tau\) restricted to the centre. Equality for every finite sum therefore gives equality on \(e\). S06 then gives \(\tau=\tau\circ T\) on all of \(M\).
For the application to an arbitrary tracial state, suppose the finite averaging construction gives, for each projection \(e\) and each dyadic \(N\), a finite average of unitary conjugates with
\[
\left\|\frac1L\sum_{j=1}^L u_j e u_j^*-T(e)\right\|\le\frac2N.
\]
Traciality gives \(\rho(u_j e u_j^*)=\rho(e)\), so applying the norm-one functional \(\rho\) yields \(|\rho(e)-\rho(T(e))|\le2/N\). Letting dyadic \(N\) increase proves agreement on projections, and S06 gives \(\rho=\rho\circ T\) on all of \(M\), without normality. S04 provides precisely the finite central thresholds used in that averaging construction, and S05 provides its matrix rank pieces. The existence and faithfulness of the centre-valued trace, the monic decomposition, projection comparison, finite complements, and the dyadic matrix averaging construction remain their respective hypotheses and providers.
## Reading
Jesse Peterson, [*Notes on operator algebras*](https://math.vanderbilt.edu/peters10/teaching/spring2020/OperatorAlgebras.pdf), dated April 27, 2020. Projection comparison and complement methods appear in Lemma 5.1.5, Proposition 5.1.9, Theorem 5.1.10 and Propositions 5.2.7–5.2.8, pp.84–90. The finite homogeneous construction is related to Proposition 5.4.2, p.95; the halving and monic methods are Lemmas 6.4.1–6.4.3 and Proposition 6.4.4, pp.102–103. The approximate center-trace method is Lemmas 6.4.8–6.4.9 and Theorem 6.4.10, pp.104–106. Here LC1 constructs a finite uniform comparison constant before its infimum is taken. CN proves the positive-map norm estimate for arbitrary elements; NA1 gives both different quadratic products explicitly.
The earlier complete proof providers are [Regular-group operator foundations, H00–H03, T04a and CS01–CS06](regular-group-operator-foundations.md#h00), Infinite tensor products, F01–F08, and [Hypertraces, GNS, COMPACT, REP and SUP](hypertraces-finite-injectivity.md#state-quotient). NC uses only the abelian faithful-trace instance of REP, whose proof precedes every center-valued trace application. The three diagrams are embedded as editable SVG in this lesson source.