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

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

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

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

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,

For finite $F$, $v_F=\sum_{i\in F}v_i$ therefore satisfies

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

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

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

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

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,

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

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

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

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

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

and $w_2$ with

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

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 $\varepsilon>0$,

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

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

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

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

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

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

Because $a\perp b$ and $v=va$, $vb=0$ and $bv^*=0$. Consequently

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

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

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 $s<t$. Indeed $(h-te)_+\le(h-se)_+$ by continuous calculus; a vector in the kernel of the latter has zero quadratic value for the former, hence is in its kernel by H02's positive-form Cauchy–Schwarz argument.

The positive and negative parts of $h-te$ annihilate each other. The negative part therefore vanishes on the range of $E_t$, and the positive part vanishes on its complementary range. Consequently

Choose $C>\|h\|$ and a finite grid $-C=t_0<t_1<\cdots<t_N=C$ with mesh at most $\delta$. Then $E_{t_0}=e$, because $h+Ce$ is positive invertible by F06, and $E_{t_N}=0$. The differences $d_k=E_{t_k}-E_{t_{k+1}}$ are mutually orthogonal projections summing to $e$. They reduce $h$, and (D01.1) gives

Orthogonal decomposition of quadratic forms therefore gives

This proves uniform approximation of every self-adjoint element by finite linear combinations of projections in its corner. If every projection of a corner is central in that corner, then (D01.2), norm closure of the center, and the real/imaginary decomposition show that corner is abelian. Zero corners are included directly.

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

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

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

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 $e-p_0<e$. This contradicts finiteness of $e\le1$ by C03. Consequently every such family is finite. This is an arbitrary-cardinality argument; it does not assume countable decomposition of $M$.

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 $c(r)<c$, and

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

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

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

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,

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.