Finite models of a von Neumann algebra

Written by GPT-6.1 Sol (OpenAI), Ultra, September 2026. Self-checked by the writing AI. New original text: public domain (CC0).

Norm approximation of a C*-algebra controls its minimal and maximal tensor norms. A von Neumann algebra has a weaker, representation-sensitive approximation topology: bounded operators can converge through their values on vectors and normal functionals. Finite completely positive models in that topology lead to semidiscreteness and to injectivity.

We will prove the equivalence between those finite models, a norm bound for multiplication by the commutant, and norm approximation in the predual. We will also prove that these conditions give completely positive extension into the von Neumann algebra. The converse implication from injectivity is proved for finite and semifinite algebras in Hypertraces and finite injective algebras, then for every von Neumann algebra in Averaging, crossed products, and injectivity.

Prerequisites are Completely positive finite models, Tensor positivity and nuclearity, and The positive cone of a standard representation. In particular, we use the exact-marginal correction and the GNS dominated-functional theorem. We also use the predual of a von Neumann algebra, normal maps, the Schwarz inequality for completely positive contractions, and ultraweak compactness. For the norm-one projection criterion we import exactly the positivity and bimodularity conclusions of Tomiyama's Theorem 1, printed pp.608–609: a norm-one linear projection from a unital C*-algebra onto a C*-subalgebra is positive and satisfies $E(axb)=aE(x)b$ for all $a,b$ in its range. Corollary 4.3 proves complete positivity from these conclusions by a matrix argument. The underlying approximation results are due to Effros and Lance; the extension and compactness methods used in Section 4 are proved in the freely readable Arveson article cited below.

We use one precise normal-representation comparison: if $\pi_1$ and $\pi_2$ are faithful normal representations of $M$, then $\pi_2$ is unitarily equivalent to the restriction of $\pi_1\otimes1_K$ to $p(H_1\otimes K)$, for some Hilbert space $K$ and projection $p\in(\pi_1(M)\otimes1_K)'$. The commutant on the restricted space is $p(\pi_1(M)'\bar\otimes B(K))p$. The complete trace-class, dominated-form and cyclic-decomposition proof is given in Normal representations inside standard amplifications; it uses no injectivity or semidiscreteness theorem and imposes no separability assumption.

No factor, separability, or faithful normal state hypothesis is imposed. We assume the algebra is nonzero; the zero algebra satisfies the approximation statements trivially. Inner products are linear in the second variable.

1. Which topology should the models approximate?

A von Neumann algebra $M$ is semidiscrete if there is a net of cpc maps

with $S_i$ normal and $T_iS_i(x)\to x$ ultraweakly for every $x\in M$. Since the middle algebra is finite-dimensional, $T_i$ is automatically normal.

The use of normal recording maps matters when we pass to preduals. An arbitrary completely positive map $M\to M_n$ can have singular coordinate functionals, which cannot define a map from $M_n^*$ into $M_*$.

Lemma 1.1. Suppose cpc maps $\Phi_i:M\to M$ converge pointwise ultraweakly to the identity. Then they converge pointwise in the $\sigma$-strong* topology.

Proof. For a cpc map $\Phi$, a contractive Stinespring operator gives $\Phi(x)^*\Phi(x)=V^*\pi(x)^*VV^*\pi(x)V\le V^*\pi(x^*x)V=\Phi(x^*x)$. Thus, for every normal positive functional $f$, the Schwarz inequality gives

The functionals $a\mapsto f(x^*a)$ are normal, so ultraweak convergence applies to the middle term. Apply the same argument to $x^*$; positive maps preserve adjoints. The seminorms $a\mapsto\{f(a^*a)+f(aa^*)\}^{1/2}$, for normal positive $f$, define the $\sigma$-strong* topology. $\square$

The special role of the identity limit is visible in this proof: the positive term $\Phi_i(x^*x)$ converges to exactly the product needed for the limiting vector estimate. An arbitrary ultraweak limit of cpc maps need not give strong convergence.

2. Multiplication by the commutant

Represent $M$ in standard form on $H$, and put $N=M'$. The commuting actions give an algebraic *-homomorphism

It always extends to the maximal tensor product. The relevant extra assertion is

Theorem 2.1. A von Neumann algebra is semidiscrete if and only if (1) holds in one faithful normal representation. In that case (1) holds in every faithful normal concrete representation. The finite models may be chosen with unital recording maps.

Proof. Suppose first that $\Phi_i=T_iS_i$ is a semidiscrete approximation. In any faithful normal representation $M\subset B(K)$, let $y_j\in M'$. For each $i$, the map

is a completely positive contraction on the maximal tensor product. This follows from the commuting Stinespring construction in Completely positive finite models: the commutant action lifts to the dilation of $T_i$, and compression gives the displayed product map. Matrix algebras have a unique C*-tensor norm, so the map is contractive for the minimal norm as well. Compose it with the minimal tensor map $S_i\otimes\operatorname{id}_{M'}$. We obtain

The left-hand operators converge ultraweakly to $\sum_jx_jy_j$. An operator-norm ball is ultraweakly closed, proving (1).

Conversely suppose (1) holds in a faithful normal representation. Lemma 2.2 below transfers the inequality to a standard representation; use that representation for the rest of the proof. Fix finitely many normal positive functionals $f_1,\ldots,f_r$ on $M$, and put $w=\sum_jf_j$. Let $\xi_w$ be its standard-cone vector. The functional

is positive on the maximal tensor product. By (1) it is bounded on the minimal tensor product. Its associated map is

Approximate $\Omega$ weak* by convex combinations of vector functionals whose vectors are finite sums of elementary tensors in the faithful spatial representation on $H\otimes H$. The factorization of such a vector functional from Tensor positivity and nuclearity has recording map

It is normal on $M$, because every coordinate is a normal vector functional. Its reconstruction map has values in $N_*$. The associated maps converge pointwise weakly in $N_*$. Convexity and Hahn–Banach on each finite product of preduals give pointwise norm approximation there.

Apply exact-marginal correction to these factorizations, with marginal $\psi=\theta(1)=\omega_{\xi_w}|_N$. Its recording maps remain normal. To check this last point, finite block sums and multiplication by fixed scalar matrices preserve normality, and the added scalar block can use a normal state of $M$. Such a state exists on every nonzero von Neumann algebra: normalize any nonzero normal positive functional. The normalization at the finite-dimensional support also preserves normality. We obtain a normal ucp $S:M\to M_n$ and a completely positive $T:M_n\to N_*$, with $T(1)=\psi$, whose composite meets any specified finite set of norm tests against $\theta$.

Let $e\in M$ be the projection onto $\overline{N\xi_w}$. The dominated-functional inverse converts $T$ into a ucp map $T':M_n\to eMe$, whose unit is $e$. As a map into $M$, it is cpc. Each $f_j\le w$ has a GNS Radon–Nikodym contraction $b_j\in N$ with $f_j(x)=\langle b_j\xi_w,xb_j\xi_w\rangle$. Consequently,

If all the selected functionals are zero, use the normal-state recording map and zero reconstruction map instead. Every normal functional is a linear combination of four positive normal functionals. Thus, by directing finite sets of operators, finite sets of normal functional tests and positive tolerances, these cpc factorizations approximate the identity pointwise ultraweakly. Their recording maps are normal and unital. This proves semidiscreteness. The first half of the proof gives (1) in every faithful normal representation. $\square$

This proof uses one normal positive functional at a time to organize finitely many tests. It does not replace an arbitrary von Neumann algebra by a countably decomposable one.

Lemma 2.2 (Transfer between normal representations). Inequality (1) passes to Hilbert-space amplifications and to faithful commutant restrictions. Consequently it passes between any two faithful normal representations.

Proof. Suppose $M\subset B(H)$ satisfies (1), so its multiplication map extends to a *-homomorphism on $M\otimes_{\min}M'$. Amplify the representation to $H\otimes K$. The commutant is $M'\bar\otimes B(K)$.

For a finite-dimensional subspace $F\subset K$, compression

is cpc. If $z=\sum_jx_j\otimes y_j$, this compression followed by multiplication is precisely the restriction to $H\otimes F$ of the compressed operator $\sum_j(x_j\otimes1)y_j$. On that finite corner, multiplication is the matrix amplification of the original minimal-tensor *-homomorphism, after permuting the tensor factors. It is contractive. CP tensor functoriality for the compression therefore gives

As $F$ increases, these compressions converge strongly to the original operator. Their uniform norm bound proves (1) for the amplification, with no restriction on the dimension of $K$.

Next take $p\in M'\bar\otimes B(K)$ such that the restriction of $M\otimes1$ to $p(H\otimes K)$ is faithful. The commutant of that restriction is $p(M'\bar\otimes B(K))p$. Inclusion of this corner into the larger commutant preserves the minimal tensor norm. The product operator on the restricted space is the compression by $p$ of the corresponding product in the amplification. Hence (1) passes to the restriction. Apply the complete normal-representation construction linked in the prerequisites to obtain the final assertion. $\square$

3. The predual formulation uses dual matrix norms

The Banach predual $M_*$ inherits the dual matrix order from $M^*$. The middle space in a contractive predual factorization is $M_n^*$, with trace norm, rather than $M_n$ with operator norm.

Theorem 3.1. The following are equivalent:

1. $M$ is semidiscrete.
2. The identity of $M_*$ is approximated pointwise in norm by cpc factorizations
   $$M_*\xrightarrow{\gamma_i}M_{n_i}^* \xrightarrow{\delta_i}M_*.$$

Complete positivity in 2 uses the dual matrix orders, and contractivity uses the Banach norms.

Proof. Adjoint a normal cpc factorization $\Phi_i=T_iS_i$ on $M$. Its preadjoint is

Preadjoints preserve norms and complete positivity in the dual orders. Ultraweak convergence of $\Phi_i(x)$ to $x$ says that $(\Phi_i)_*(f)\to f$ weakly in $M_*$, for every $f\in M_*$.

The set of such composites is convex. Indeed, take finite convex combinations of the primal factorizations using a block diagonal recording map and a convex sum reconstruction map. Both remain cpc and the recording map remains normal. Pass to the preadjoints. Applying Hahn–Banach to the image in $(M_*)^r$, for each finite list of functionals, changes weak approximation into norm approximation. This proves 2.

Conversely, adjoint a factorization in 2. The adjoint of $\delta_i$ is a normal cpc recording map $M\to M_{n_i}$, and the adjoint of $\gamma_i$ is a cpc reconstruction map $M_{n_i}\to M$. Normality of the first map follows precisely because $\delta_i$ has range in the predual. For $x\in M$, $f\in M_*$,

Thus these maps prove semidiscreteness. $\square$

Under separability of $M_*$, norm approximation in 2 can be arranged as a sequence by the usual dense-list argument with a uniform contractive bound. Without separability, the finite-set formulation gives a net. Lemma 1.1 also shows that the primal approximations converge in the $\sigma$-strong* topology.

4. The finite models give an extension property

Call a von Neumann algebra $M$ injective if every completely positive map from an operator system $S\subset C$, where $C$ is a unital C*-algebra, to $M$ extends completely positively to $C$. The extension retains the value at the unit, hence retains the norm; unital maps have unital extensions. No normality of the extension is requested.

Proposition 4.1. In a faithful normal representation $M\subset B(H)$, injectivity is equivalent to the existence of a ucp retraction $E:B(H)\to M$. This criterion is independent of the faithful normal representation.

Proof. Extend the identity map on the operator system $M$ to $B(H)$ by injectivity. The extension takes the unit to the unit, has range in $M$, and fixes $M$, giving the retraction.

Conversely, given $E$, first extend a completely positive $\phi:S\to M\subset B(H)$ by Arveson's theorem to a map $\Phi:C\to B(H)$. Then $E\Phi$ has range in $M$ and agrees with $\phi$ on $S$. Its unit value is unchanged because $E$ fixes $M$. Hence it has the same norm and is unital when $\phi$ is unital. This is an abstract property of $M$, proving the representation assertion. $\square$

Theorem 4.2. Every semidiscrete von Neumann algebra is injective.

Proof. Represent $M\subset B(H)$ faithfully and normally, and take its normal cpc finite models $T_iS_i$. Arveson's theorem extends each $S_i$ to a cpc map $\widehat S_i:B(H)\to M_{n_i}$; its value at the unit is preserved. The composite $E_i=T_i\widehat S_i:B(H)\to M$ is cpc.

For each $a\in B(H)$, the values $E_i(a)$ lie in the ultraweakly compact ball of radius $\|a\|$ in $M$. A subnet converges ultraweakly at every $a$, by compactness of the product of these balls. The limit $E$ is linear and completely positive, since matrix positive cones are ultraweakly closed. For $x\in M$, $E_i(x)=T_iS_i(x)\to x$, so $E(x)=x$. In particular $E(1)=1$. It is a ucp retraction, and Proposition 4.1 gives injectivity. $\square$

The extension $\widehat S_i$ and the limiting retraction need not be normal. Ultraweak compactness gives their existence but does not assert ultraweak continuity. This is consistent with the definition of injectivity.

Corollary 4.3 (Norm-one projection criterion). A nonzero von Neumann algebra $M\subset B(H)$ is injective if and only if it is the range of a bounded linear projection of norm one on $B(H)$.

Proof. Injectivity gives the ucp retraction of Proposition 4.1. It is a projection of norm one because it is contractive and fixes the unit.

Conversely, let $E:B(H)\to M$ be a norm-one projection. It fixes $M$, and Tomiyama's Theorem 1 gives positivity and $M$-bimodularity. We derive complete positivity explicitly. For $X=[x_{ij}]\ge0$ in $M_n(B(H))$, put $Y=[E(x_{ij})]$. Positivity of $E$ makes $Y$ self-adjoint. For every column $b=(b_1,\ldots,b_n)^{\mathsf T}$ with entries in $M$,

Here the last input is positive because it equals $b^*Xb$.

To see that these tests imply $Y\ge0$, let $R=Y_-$ be its negative part and let $c_k$ be the $k$-th column of $R^{1/2}\in M_n(M)$. Functional calculus gives

Equation (2) makes this nonnegative, whereas $(R^2)_{kk}\ge0$. Thus $0=(R^2)_{kk}=\sum_iR_{ik}^*R_{ik}$ for every $k$, so every entry of $R$ vanishes. Hence $Y\ge0$. This holds for every $n$, proving that $E$ is completely positive. It fixes the shared unit, so is ucp. Proposition 4.1 gives injectivity. No normality of the projection follows or is needed. $\square$

5. Exercises with solutions

Exercise 1 (A normal finite model for all bounded operators; intermediate). On an arbitrary Hilbert space $H$, let $p_K$ range over the finite-dimensional subspace projections. Show that compression $B(H)\to B(K)$, followed by inclusion $B(K)\to B(H)$, proves semidiscreteness of $B(H)$.

Solution. Both maps are cpc and normal, and the composite is $x\mapsto p_Kxp_K$. The net $p_K$ converges strongly to one. On bounded operators this gives strong convergence of the composites to $x$, and applying it to $x^*$ gives strong* convergence. The same conclusion holds in the $\sigma$-strong* topology: normal positive functionals are sums of vector functionals, and the uniform norm bound controls the tails of such sums. Thus the algebra is semidiscrete. When $H$ is nonseparable, the finite subspaces are directed by inclusion rather than a sequence.

Exercise 2 (Why the limit map matters; intermediate). Give cpc maps converging pointwise ultraweakly to a map other than the identity, but failing pointwise strong convergence.

Solution. Let $H=\mathbb Ce_0\oplus\mathbb Cf\oplus\ell^2(\mathbb N)$, with all the indicated vectors orthonormal. Let $u_n$ be the self-adjoint unitary exchanging $e_0$ and the $n$-th basis vector $e_n$ of the last summand and fixing the orthogonal complement. The automorphisms $\Phi_n(x)=u_nxu_n$ are ucp. Compactness of the product of the ultraweak operator-norm balls gives a subnet converging pointwise ultraweakly to a ucp map $\Phi$.

For $x=|e_0\rangle\langle f|$, the images are $|e_n\rangle\langle f|$. They converge weakly as operators to zero and are uniformly bounded, hence converge ultraweakly to zero. Thus $\Phi(x)=0$. But $\|\Phi_n(x)f\|=\|e_n\|=1$, so the subnet cannot converge strongly on this $x$. The limit is not the identity: the images of the rank-one projection $p_0$ are $p_n$, which converge ultraweakly to zero, giving $\Phi(p_0)=0$.

Exercise 3 (Normality belongs to the predual; intermediate). Why does an arbitrary bounded map $S:M\to M_n$ not always give a preadjoint $M_n^*\to M_*$?

Solution. Its Banach adjoint always maps $M_n^*$ into $M^*$. It has range in $M_*$ exactly when each scalar coordinate functional of $S$ is normal, which is equivalent to normality of $S$. For example, a singular state on an infinite-dimensional von Neumann algebra, viewed as a map to $\mathbb C$, has an adjoint sending the scalar unit to that singular state, outside $M_*$.

Exercise 4 (The unit of a corner; introductory). In Theorem 2.1, why is $T'$ cpc as a map into $M$ even though it is unital only as a map into $eMe$?

Solution. Its value at $1_n$ is $e$, a projection of norm at most one. For a completely positive map from a unital algebra, its norm is the norm of its value at the unit. Inclusion of the corner preserves positivity at every matrix size. Hence it is cpc in $M$.

Exercise 5 (Amplifying weak convergence; intermediate). In Lemma 1.1, show that $a\mapsto f(x^*a)$ is a normal functional whenever $f\in M_*$ and $x\in M$.

Solution. Left multiplication by a fixed element is ultraweakly continuous in a von Neumann algebra. Composing it with the normal functional $f$ gives a normal functional. Equivalently, realize $f$ as an absolutely summable series of vector coefficients; replace the first vector in each coefficient by its image under $x$. The bounded multiplier preserves absolute summability.

Exercise 6 (Keep the extension's unit value; intermediate). In Theorem 4.2, do the cpc maps $E_i$ have to be unital? Why is their limit unital?

Solution. The reconstruction map can have unit value a proper projection or another positive contraction, so the composite need not be unital. On the particular input $1\in M$, however, $E_i(1)=T_iS_i(1)\to1$ ultraweakly. The limit therefore satisfies $E(1)=1$. It is the approximation to the identity on the subalgebra that gives unitality at the limit.

References

Jun Tomiyama, On the projection of norm one in W*-algebras, Proceedings of the Japan Academy 33, no.10 (1957), 608–612. Theorem 1 and its complete proof, printed pp.608–609 (PDF pp.1–2), give positivity, bimodularity, and the Schwarz inequality for a norm-one projection from a unital C*-algebra onto a C*-subalgebra. Its proof first reduces to the biduals and then uses range projections and norm estimates to obtain bimodularity; the final positive-square computation gives Schwarz. Corollary 4.3 uses precisely positivity and bimodularity, and supplies the additional matrix argument needed for complete positivity. In its concrete von Neumann algebra setting no separability, finite-dimensionality, or normality of the projection is assumed.

William B. Arveson, Subalgebras of C*-algebras, Acta Mathematica 123 (1969), 141–224. Theorem 1.1.1 gives the dilation underlying Lemma 1.1’s Schwarz estimate. Theorem 1.2.3 and its proof give the completely positive extension and bounded compactness methods expanded in Completely positive finite models. Section 4 applies those methods to construct the retraction explicitly, for arbitrary Hilbert spaces and von Neumann algebras. It proves semidiscreteness implies injectivity without a separability or faithful normal state assumption.

The commutant characterization in Section 2 also uses the tensor-positivity lesson's exact-marginal correction and its explicit proof of both matrix-order directions for dominated functionals. Arveson's Lemma 1.4.1 and Theorem 1.4.2, printed pp.159–160, supply the corresponding dilation-commutant order method. The finite models in Section 2 retain normal recording maps through every block sum, scalar addition and support normalization; Section 3 then uses their actual preadjoints with dual matrix norms.

Huzihiro Araki, Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon–Nikodym theorem with a chain rule, Pacific Journal of Mathematics 50 (1974), 309–354, Theorem 4(8), printed p.332, gives the cone-vector estimate used in marginal correction. Uffe Haagerup, The standard form of von Neumann algebras, Mathematica Scandinavica 37 (1975), 271–283, Lemma 2.10, carries that estimate to arbitrary von Neumann algebras by support corners. These inputs do not require a faithful normal state on the whole algebra. The normal representation comparison and Tomiyama's norm-one projection theorem retain their exact stated prerequisites.