Central traces and AFD finite algebras

AI-generated exposition by OpenAI GPT models. CC0-1.0 to the extent of rights held.

An AFD finite algebra need not be a factor. Its matrix blocks can have different dimensions on different parts of the center. We will cut those dimensions to dyadic values, align an old matrix algebra exactly, and prove that a separable AFD algebra of type II₁ is the tensor product of its center with the tracial infinite product of $M_2$.

We use the full center-valued trace and projection proofs: existence, faithfulness and normal uniqueness of $T:M\to Z(M)$, comparison by this trace, type II halving, and the finite type I central structure. Local approximation and the hyperfinite finite factor proves the projection-rotation lemma in arbitrary finite algebras, matrix coordinates, and bounded-ball completeness for the trace $2$-norm. Finite trace expectations, TL3, constructs the normal, ucp, bimodular, trace-preserving expectation onto every unital subalgebra and identifies it with the trace Hilbert-space orthogonal projection.

In the central-dimension, matrix-enlargement and tensor-product arguments, $M$ is of type II₁ and has a faithful normal tracial state $\tau=\rho T$, where $\rho$ is a faithful normal state on $Z=Z(M)$. Set $\|x\|_2=\tau(x^*x)^{1/2}$. The dimension and enlargement arguments require neither factoriality nor separability; Theorem 3.1 assumes separable predual. The directed averaging assertion of Theorem 4.1 allows an arbitrary von Neumann algebra. On bounded sets this norm gives sigma-strong* convergence, by the trace-density estimate in the preceding finite-factor lesson. Local AFD and finite-set $2$-norm approximation agree: expectations onto the finite-dimensional approximating algebra supply bounded approximants.

The center-valued trace is precisely the trace expectation onto $Z$. Indeed its central bimodularity and $\tau=\rho T$ give $\tau(zT(x))=\tau(zx)$ for every central $z$. The same pairing characterizes $E_Z(x)$. Their difference is central; pairing with its adjoint and using faithfulness of $\tau$ makes the difference zero. Thus $T=E_Z$, and in particular $T$ is contractive in the trace $2$-norm, as used below.

The finite-group commutant argument first explains how an increasing finite-dimensional family gives injectivity. Central dimension cuts then produce finite matrix models. Transporting a repaired corner retains an old matrix factor exactly, and the resulting nested factors give the center tensor-product decomposition.

Directed finite averages and injectivity

Theorem 4.1. A von Neumann algebra generated by an increasing directed family of finite-dimensional unital algebras is injective. Every locally AFD algebra with separable predual has such a generating sequence, and is therefore injective.

Proof. The directed averaging argument uses no separability. Its local-to-sequential consequence uses Theorem 3.1, proved after the central-dimension and matrix-enlargement constructions below. For the first assertion, in standard form average over the compact unitary group of each finite algebra:

These maps may be computed by finite averages. For each full matrix block of $D_i$, take its cyclic shift and diagonal clock unitaries, extended by the identity off that block, and also the central sign which is minus one on that block and one on the others. They generate a finite group: within each block clock and shift commute up to a root of unity, and the block signs commute with all generators. The algebra generated by this group is exactly $D_i$. Averaging conjugation over the finite group is therefore normal, unital and completely positive, fixes $D_i'$, and has range in $D_i'$. The central signs kill cross-block terms; the clock and shift average is the normalized matrix trace on each matrix leg. This is also the compact-unitary average in (12), so no additional integration construction is needed for these maps.

These are ucp maps onto $D_i'$. Product ultraweak compactness gives a pointwise convergent subnet. For each fixed $j$, its tail values lie in $D_j'$, so the limit $F$ takes values in $\bigcap_jD_j'=M'$. It fixes $M'$. The map $b\mapsto JF(JbJ)J$ is a linear unital completely positive retraction onto $M$: conjugation by $J$ twice is linear, and at each matrix size it carries positive operator matrices through a conjugate Hilbert-space identification. It fixes $M$ because $JMJ=M'$. The norm-one projection criterion gives injectivity. This is also the exact directed permanence result already taught in the averaging lesson; no separability is needed here.

For the second assertion, decompose the identity into its finite type I, finite type II and properly infinite central parts. Local AFD passes to a central part by multiplying the approximating finite algebra by that central projection. The properly infinite part has a dyadic generating sequence by the preceding lesson. The finite type II part has the sequence $D_n$ constructed in Theorem 3.1. A finite type I algebra is a countable central product of homogeneous algebras $Z_r\bar\otimes M_r$. Each separable abelian $Z_r$ is generated by an increasing sequence of finite projection partitions. Tensor these finite central algebras with $M_r$, retain the first $n$ homogeneous summands, and add the scalar identity on the remaining central tail. These are increasing finite-dimensional algebras generating the type I part. Combining the three sequences by direct sums gives a generating sequence for $M$. Apply the first assertion. $\square$

The finite type I structure used here follows from the abelian-projection construction in FCT3. On the central support of an abelian projection, that construction splits the identity into central parts, each filled by finitely many equivalent abelian projections. Their connecting partial isometries are matrix units; the corner-center identity identifies every matrix entry with the central algebra. Thus each part is a full matrix algebra over an abelian center. A maximal orthogonal family of such central supports exhausts the finite type I unit. The faithful normal state available under separable predual makes this family countable, because only finitely many disjoint projections can have state value above any fixed positive threshold. Group the parts of equal matrix size and sum their corresponding matrix units over the orthogonal central parts. This gives the asserted homogeneous central product. Refining its finite central partitions and then including one more homogeneous summand preserves the preceding finite algebra: its old scalar tail is the sum of the new summand's identity and the new scalar tail. Hence the type I sequence is genuinely increasing and generates the whole part.

The finite injective factor case is treated in Unitary couplings and finite injective factors.

Central dimensions and dyadic rounding

Lemma 1.1. If $e$ is a projection and $a\in Z_+$ satisfies $a\le T(e)$, there is $g\le e$ with $T(g)=a$.

Proof. On the central support of $e$, put $r=a/T(e)$ by measurable functional calculus, taking zero off that support. It is a bounded central element with $0\le r\le1$. Repeated halving gives orthogonal pieces $p_j\le e$, $j\ge1$, with

Indeed retain one half at each stage and halve the remaining half. The trace of the decreasing remainder tends to zero; faithfulness and normality of $T$ make its limiting projection zero. Express the central function $r$ in binary. Its digits are central projections $z_j$, with $r=\sum_{j\ge1}2^{-j}z_j$; at $r=1$ take all digits one. The strong sum $g=\sum_jz_jp_j$ is a subprojection of $e$, and normality gives $T(g)=rT(e)=a$. $\square$

In particular every projection can be split into a prescribed finite list of nonnegative central dimensions summing to its dimension. Apply the lemma successively to the remaining projection. Projections with equal center-valued trace are equivalent by the declared finite comparison theorem.

The central ratio and binary digits in this argument also have a bounded spectral construction. Put $t=T(e)$. The commuting positive elements $a(t+1/n)^{-1}$ increase and are bounded by one; their strong limit is a central positive contraction $r$, supported on the support of $t$. Multiplication by $t$ gives $rt=a$, since the positive difference $a-at(t+1/n)^{-1}$ is bounded by $1/n$. Dyadic interval projections of $r$ give its binary digits by finite spectral step functions, with all digits one on the spectral projection where $r=1$. Their increasing binary partial sums have error at most $2^{-n}$ in operator norm. This supplies the ratio and digit operations without a separate unbounded measurable-field theorem.

Lemma 1.2. If $D\subset M$ is a finite-dimensional unital algebra and $\eta>0$, there exist a unital dyadic matrix factor $Q\cong M_{2^p}$ and a finite central partition $z_1,\ldots,z_s$ such that, for $P=\sum_\ell Qz_\ell$,

The exponent $p$ can exceed any specified integer.

Proof. Write $D$'s orthogonal matrix systems as $f_{ij}^{(k)}$, with sizes $d_k$, and set $t_k=T(f_{11}^{(k)})$. Put $N=2^p$, and round each central dimension downward:

Choose $g_k\le f_{11}^{(k)}$ with $T(g_k)=\alpha_k$, by Lemma 1.1, and transport it through its block:

The integer-valued central functions $r_k=\lfloor Nt_k\rfloor$ take finitely many values. Their joint level sets give a finite central partition $(z_\ell)$. On one such part, split $g_kz_\ell$ into $r_k$ pieces of center-valued trace $z_\ell/N$, and transport each piece through the $d_k$-block. When $r_k=0$ that block contributes no piece. The complement has trace

and Lemma 1.1 splits it into that many further pieces. There are now exactly $N$ equivalent diagonals on $z_\ell$. Complete them to an $N$-matrix system while respecting the transported block connections: connect a reference diagonal to the first diagonal of each prescribed block, then use its existing matrix units for the other diagonals. Products of the connecting partial isometries give all the required units.

Do this on each central part, and sum corresponding matrix units over $\ell$. Their sums form a global unital $N$-matrix system with factor span $Q$. Although the ordering of the old blocks can vary with $\ell$, every $\widetilde f_{ij}^{(k)}$ lies in $P=\sum Qz_\ell$. If $x=\sum\lambda_{ij}^{(k)}f_{ij}^{(k)}$, then $|\lambda_{ij}^{(k)}|\le\|x\|$. With $s=\sum_kd_k^2$, its corresponding truncated sum in $P$ differs by at most $sN^{-1/2}\|x\|$. The nearest-point property of $E_P$ proves (2) when $p$ is sufficiently large. $\square$

The central partition in this construction is essential. The ranks $r_k$ generally depend on the central variable; a single scalar trace does not record that dependence.

Keeping a matrix factor through its corner

In a locally AFD $M$, every finite-dimensional linear subspace $V$ admits approximation of its entire operator-norm unit ball by an algebra $P$ as in Lemma 1.2. Choose a basis of $V$; the sum of the absolute values of its coordinates is bounded on that unit ball. Approximate the basis by a finite-dimensional algebra, then apply Lemma 1.2 to that algebra. The coordinate bound makes the approximation uniform. Expectations preserve the norm bounds.

Lemma 2.1. Suppose $M$ is locally AFD, $A\cong M_d$ is a unital subfactor with $d=2^n$, and $V\subset M$ is finite dimensional. Given $\varepsilon>0$, there are a dyadic factor $B\supset A$ and a finite central partition $(z_\ell)$ such that

The size of $B$ may be made strictly larger than that of $A$.

Proof by repaired-corner transport. Put $e=e_{11}$ and

Choose $P=\sum_\ell Qz_\ell$, with $Q\cong M_N$, $N=2^p$, $p>n$, so that

The uniform finite-dimensional approximation preceding the statement supplies this choice, with $\delta>0$ to be fixed below.

We first repair the unit of the desired corner. For $h=E_P(e)$, positivity and contractivity give $0\le h\le1$ and $\|h-e\|_2\le\delta$. The identity

gives $\|h-h^2\|_2\le3\delta$. Let $f=1_{[1/2,1]}(h)\in P$. The scalar bound

and the triangle inequality give $\|f-e\|_2\le7\delta$. Since $T=E_Z$ is a trace-Hilbert-space contraction and $T(e)=1/d$,

On every central part $z_\ell$, enlarge or shrink $f z_\ell$ to a comparable projection $g z_\ell$ of rank $N/d$ in $Pz_\ell\cong M_N$. This integer rank is available because $p>n$. The resulting projection $g\in P$ satisfies

For the middle equality, $f-g$ is, up to sign on each part, a projection of the rank difference. The inequality is Cauchy--Schwarz for the state $\rho$, followed by the preceding $L^2$ bound. Equal center-valued traces and the finite projection-rotation lemma give a unitary $u\in M$ with

For $a=eae\in W$, define

We have $\|gE_P(a)g\|\le\|a\|$. Also

where

The tracial multiplication bounds and (CT1)--(CT3) therefore give

Here conjugation by $u$ adds at most $2\|u-1\|_2\|a\|$. In particular $K_\delta\to0$ as $\delta\downarrow0$.

Each central part of $D_0$ is a matrix algebra of size $N/d$, with unit $ez_\ell$. Choose matrix units on each part and sum corresponding units over $\ell$. Their global span is a matrix factor $C_0\cong M_{N/d}$, with unit $e$, and $D_0=\sum_\ell C_0z_\ell$. Define

Matrix-unit multiplication shows that $\theta$ is a unital $*$-homomorphism into $A'\cap M$: both $\theta(c)e_{ij}$ and $e_{ij}\theta(c)$ equal $e_{i1}ce_{1j}$, and $e\theta(c)e=c$. It is injective, preserves the unit as $\theta(e)=1$, and commutes with central multiplication. Consequently

contains $A$ exactly. For completeness, if $c_{rs}$ are the units of $C_0$, then $e_{ij}\theta(c_{rs})$ form a full $N$-matrix system: multiplication and adjoints have the two Kronecker index rules, their diagonal sum is one, and their diagonals $e_{i1}c_{rr}e_{1i}$ are nonzero. This proves the claimed factor size without an abstract tensor-injectivity assumption.

For $x\in V$, put $a_{ij}=e_{1i}xe_{j1}\in W$ and

The equality follows directly from (CT5); membership follows from $b(a_{ij})\in D_0$. The matrix coefficients are orthogonal for the tracial $2$-inner product. Indeed cyclically moving the outside units in the pairing of the $(i,j)$ and $(k,l)$ terms gives the factors $\delta_{ik}\delta_{jl}$, while a diagonal pairing equals $\|a_{ij}-b(a_{ij})\|_2^2$. Thus

We used $\|a_{ij}\|\le\|x\|$. Choose $\delta$ with $dK_\delta<\varepsilon$. The nearest-point property of $E_{\widetilde P}$ proves (5). Since $p>n$, $B$ is strictly larger than $A$.

The construction uses the local AFD approximation of $W$ in the ambient algebra, followed by explicit finite rank repair. It does not assume that $eMe$ is AFD or amenable. Orthogonal trace-Hilbert-space matrix coordinates also occur in Anantharaman--Popa, Theorem 11.2.2; their factor corner induction uses amenability, whereas the repaired corner above is built directly. $\square$

Second proof, by aligning full matrix units. Fix matrix units $e_{ij}$ of $A$, put $e=e_{11}$, and set $W=\mathbb Ce+\sum_{i,j}e_{1i}Ve_{j1}$. The preceding uniformity gives $P=\sum Qz_\ell$, with $Q\cong M_{2^p}$, $p>n$, such that $\|a-E_P(a)\|_2\le\delta\|a\|$ on $W$. Put $h=E_P(e)$. Then

For the last inequality use $h-h^2=(h-e)+(e-h)e+h(e-h)$ and the tracial multiplication bounds. Spectral rounding at $1/2$ gives $f=1_{[1/2,1]}(h)\in P$ and $\|f-e\|_2\le7\delta$: pointwise $|t-1_{[1/2,1]}(t)|\le2t(1-t)$.

Since $T(e)=1/d$, contraction of the center expectation gives

On each $z_\ell$, the projection $f$ has integer matrix rank. Enlarge or shrink it inside that matrix block to a comparable projection of rank $2^p/d$. Assembling these projections gives $g\in P$, with $T(g)=1/d$ and

Equal center-valued traces give $g\sim e$. The finite projection-rotation lemma from the preceding lesson gives a unitary $u\in M$ with $ugu^*=e$ and $\|u-1\|_2\le\sqrt2a(\delta)$.

In each central matrix block, split the space into $d$ equal subspaces with first projection $g z_\ell$. Summing their matrix units gives unital $d$-matrix units $w_{ij}\in P$, with $w_{11}=g$. Set

Multiplication gives $v^*v=vv^*=1$ and $vw_{ij}v^*=e_{ij}$. Thus $P'=vPv^*$ contains $A$. Also $vg=ug$. For $a=eae\in W$, put $a'=gE_P(a)g$. Then $\|a'\|\le\|a\|$, and

The first estimate uses $a=eae$ and tracial multiplication; the second uses $va'v^*=ua'u^*$ and $\|u-1\|_2$. Reconstructing $x=\sum e_{i1}(e_{1i}xe_{j1})e_{1j}$ shows that its distance to $P'$ is at most $d^2K(\delta)\|x\|$.

Each central component of $P'$ is $M_{2^p}$, and contains the fixed $M_d$ algebra $A z_\ell$. Its relative commutant is $M_{2^p/d}$. Choose matrix units in these relative commutants and sum them over $\ell$; they generate a global factor $C\cong M_{2^p/d}$ commuting with $A$. Put $B=A\vee C\cong M_{2^p}$. Matrix coordinates on each central part give $P'=\sum Bz_\ell$, and $A\subset B$ exactly. Choose $\delta$ so $d^2K(\delta)<\varepsilon$, and use the nearest-point property of its expectation. $\square$

Only $u$, the unitary aligning the first projection, must be close to one. No smallness is required of the full matrix-unit aligning unitary $v$.

Nested factors and the center tensor product

Theorem 3.1. If $M$ is a locally AFD type II₁ von Neumann algebra with separable predual, then

Proof. Choose a faithful $\rho$ on $Z$, hence $\tau=\rho T$, and a $2$-norm dense sequence $(x_j)$ in the unit ball. Separability also supplies a countable family of central projections generating $Z$: take self-adjoint generators and their rational spectral projections. Start with $A_0=\mathbb C1$. Apply Lemma 2.1 successively to the span of the first $n$ elements, with error $1/n$, retaining $A_{n-1}$ exactly and increasing the matrix size. Let $Z_n$ be the finite central algebra generated by all the partitions obtained so far and the first $n$ chosen central projections. Then

are increasing finite-dimensional unital algebras, and $\|x_j-E_{D_n}(x_j)\|_2\le1/n$ for $j\le n$. Their bounded approximants and trace $2$-norm completeness imply $(\bigcup D_n)''=M$.

Put $R_0=(\bigcup A_n)''$. It is finite. Every normalized normal trace on $R_0$ restricts to the unique normalized trace on every $A_n$, so all such traces agree on the union and, by normality, on $R_0$. If its center were nontrivial, composing its center-valued trace with different normal center states would give different normal traces. Thus $R_0$ is a factor. Its unbounded matrix sizes rule out a finite-dimensional factor, so it is type II₁. Insert the missing dyadic tensor levels between the $A_n$'s. Their trace is the product matrix trace; the trace GNS construction identifies $R_0$ with $R$, as proved in the infinite-product and finite-factor lessons.

For $z\in Z_+$, $\tau(z)>0$, the functional $x\mapsto\tau(zx)/\tau(z)$ is a normalized normal trace on $R_0$, hence equals $\tau|_{R_0}$. By linearity, including $z=0$,

Consequently multiplication $Z\odot R_0\to M$ preserves the product trace inner product: expand the pairing of two finite tensor sums and use (11). It extends to a unitary from the product trace Hilbert space onto $L^2(M,\tau)$, since $Z$ and $R_0$ generate $M$. This unitary intertwines left multiplication by both tensor legs. Their generated von Neumann algebras are therefore normally isomorphic, proving (10). $\square$

5. Problems with complete solutions

Commutants and the tensor-product conclusion

Exercise 10. Why must the averaging proof use commutants rather than just a pointwise limit of retractions onto the $D_i$'s?

Solution. A limit of retractions onto $D_i$ fixes their union, but may be singular and therefore need not fix its ultraweak closure. Averaging instead gives ranges in the decreasing algebras $D_i'$. For each fixed index all sufficiently late values lie in its closed commutant, so the entire limiting range lies in their intersection $M'$. The limit fixes that intersection directly. Standard-form conjugation then supplies the retraction onto $M$.

Exercise 9. Prove that (11) makes multiplication isometric on trace Hilbert spaces.

Solution. For a finite sum $t=\sum_i z_i\otimes x_i$, its product satisfies $\|\sum_i z_ix_i\|_2^2=\sum_{i,j}\tau(z_i^*z_jx_i^*x_j)=\sum_{i,j}\tau(z_i^*z_j)\tau(x_i^*x_j)$. This is exactly the squared Hilbert tensor-product norm of $t$. The generation hypothesis makes the image dense, so the isometry extends onto the entire trace Hilbert space and intertwines the two left actions.

Exact matrix containment

Exercise 7. Verify that the unitary (8) aligns the old matrix units.

Solution. In $v^*v$, $e_{1i}e_{j1}=\delta_{ij}e$, and $u^*eu=g$, so the sum reduces to $\sum_iw_{i1}gw_{1i}=\sum_iw_{ii}=1$. The reverse product similarly gives $\sum_ie_{ii}=1$. Inserting $w_{rs}$ between the sums gives $e_{r1}u g u^*e_{1s}=e_{rs}$. Thus all the old units lie exactly in $vPv^*$.

Exercise 8. Why does the new factor $B$ in Lemma 2.1 have size $2^p$ even when the central partition has many pieces?

Solution. On every central piece the old algebra acts as $M_d$, and its relative commutant is $M_{2^p/d}$. Summing corresponding units of these commutants gives one global matrix factor of that size. Together with the old algebra it gives $M_d\otimes M_{2^p/d}=M_{2^p}$. The central projections remain in the larger algebra $\sum Bz_\ell$, rather than becoming extra central projections of the factor $B$ itself.

Central dimensions and finite rank repair

Exercise 1. Why do the binary halves in (1) sum to $e$?

Solution. After $n$ stages the remaining projection has center-valued trace $2^{-n}T(e)$. The remainders decrease, so their strong limit has trace zero by normality of $T$. Faithfulness makes that limiting projection zero. Subtracting the remainders from $e$ gives the increasing partial sums of the selected halves and hence the asserted sum.

Exercise 2. For $M=L^\infty[0,1]\bar\otimes R$, explain why the cut dimension $a$ may vary with the variable in $[0,1]$.

Solution. The center is $L^\infty[0,1]$, and $T(e)$ is a measurable dimension function. The binary digits of $a/T(e)$ are measurable central projections, so Lemma 1.1 chooses different halves on their corresponding measurable parts. The resulting single projection has the prescribed dimension function. This requires no choice of an independently measurable family of arbitrary fibre projections.

Exercise 3. Check the error formula (4).

Solution. The difference is $f_{i1}(f_{11}-g_k)f_{1j}$. Its adjoint times itself is $f_{j1}(f_{11}-g_k)f_{1j}$. Traciality makes its trace $\tau(f_{11}-g_k)=\rho(t_k-\alpha_k)$. The downward rounding error is less than $1/N$ as a central function, so its scalar trace is less than $1/N$.

Exercise 4. Why is the number of leftover diagonals in Lemma 1.2 an integer and nonnegative on every central part?

Solution. On that part all $r_k$ are integers, so $N-\sum d_kr_k$ is an integer. The original unital blocks satisfy $\sum d_kt_k=1$, and $r_k/N\le t_k$. Therefore $\sum d_kr_k\le N$. The residual projection's trace is exactly this nonnegative integer times $z/N$, which Lemma 1.1 can split into that many pieces.

Exercise 5. Derive (6) without replacing center-valued traces by scalar ranks.

Solution. The map $T$ is the $\tau$-preserving expectation onto $Z$, hence is the orthogonal projection in $L^2$ and contractive there. Since $T(e)=1/d$, $\|T(f)-1/d\|_2=\|T(f-e)\|_2\le\|f-e\|_2\le7\delta$. Scalar ranks enter only after restricting to the finite central matrix blocks of $P$.

Exercise 6. Explain why $\|f-g\|_2^2=\rho(|T(f)-1/d|)$.

Solution. On each central block $f$ and $g$ are comparable. The square of their difference is their positive projection difference, whose center-valued trace is the absolute difference of their center-valued traces. Sum over the orthogonal central partition and apply $\tau=\rho T$. Cauchy–Schwarz for the state $\rho$ bounds the resulting $L^1$ norm by the $L^2$ norm in (6).

References and proof scope

Claire Anantharaman and Sorin Popa, An introduction to II₁ factors, author draft. Proposition 4.1.6, p.61, gives the factor binary trace-cut argument. Propositions 9.1.6 and 9.1.8, p.142, give tracial center-expectation and projection-comparison methods. Remark 9.1.7 refers unrestricted center-trace existence to another work. The full existence, faithfulness, normal uniqueness and comparison arguments used here are the local PC/FCT constructions.

The same draft's Lemma 11.2.1 and Theorem 11.2.2, pp.185–188, develop factor dyadic approximation, exact matrix containment and trace-GNS identification. Its corner induction uses amenability. The repaired-corner proof here constructs the needed finite corner directly from local AFD and central rank repair. The orthogonal trace-Hilbert-space matrix estimate is the same elementary mechanism used in the draft's factor construction; the local proof retains varying central ranks and gives the explicit bound $dK_\delta$. The full matrix-unit-aligning proof is also included.

Theorem 10.2.4, p.162, proves the compact-unitary/commutant retraction method for the hyperfinite factor $R$. The directed proof here uses exact finite clock, shift and central-sign averages and treats finite type I, finite type II and properly infinite central parts in the local-to-sequential conclusion.

Tracial expectations are supplied by TL3 in Finite trace expectations. The arbitrary-algebra standard representation and directed injectivity argument are in Averaging, crossed products, and injectivity. The properly infinite sequence uses the complete dyadic construction.