Injective von Neumann algebras

Written by Claude Opus 5.5 (Anthropic), September 2026, revised October 2026. The September text was spot-checked by Claude Opus 5.5 in a separate session; the October revisions are self-checked by the writing AI. The proof of Theorem 7.1 was drafted by GPT-6 Astra (OpenAI) in ChatGPT web, Pro mode, and checked and adapted by Claude Opus 5.5, October 2026. Public domain (CC0).

Introduction

The Hahn–Banach theorem says that a bounded linear functional defined on a subspace extends, without increase of norm, to the whole space. In the category whose objects are unital C*-algebras and whose morphisms are unital completely positive maps, Arveson's extension theorem says the same for the target \(B(H)\): a unital completely positive map from a C*-subalgebra into \(B(H)\) extends to the whole algebra. A C*-algebra with this extension property is called injective. This lesson studies the von Neumann algebras that are injective.

For a von Neumann algebra \(M\subset B(H)\), injectivity turns out to be robust, and it can be tested in many ways. The lesson proves the following.

1. \(M\) is injective exactly when some projection of norm one maps all of \(B(H)\) onto \(M\) (Section 2). In particular this does not depend on how \(M\) is represented.
2. \(M\) is injective exactly when certain matrix inequalities with coefficients in \(B(H)\) can be solved with coefficients in \(M\) (Section 3). This criterion is the tool for measurable arguments.
3. The commutant of an injective von Neumann algebra is injective; the weak closure of an increasing union and the intersection of a decreasing family of injective von Neumann algebras are injective (Section 4).
4. If an injective subalgebra and an amenable group of unitaries normalizing it generate a von Neumann algebra, that algebra is injective (Section 5). Consequences: every abelian, every type I and every approximately finite-dimensional von Neumann algebra is injective. The same holds for the von Neumann algebras generated by representations of amenable locally compact groups, their commutants, crossed products by amenable groups and the group measure space construction for actions of amenable groups. Conversely, the von Neumann algebra of a discrete group is injective only when the group is amenable, so the von Neumann algebra of the free group on two generators is not injective.
5. A direct integral of von Neumann algebras is injective exactly when almost all of its fibres are injective (Section 6).
6. Every strongly continuous unitary representation of a connected locally compact group on a separable Hilbert space generates an injective von Neumann algebra, and so does every such representation of a locally compact group \(G\) whose quotient \(G/G_0\) by the identity component is amenable (Section 7). The proof reduces to Lie groups by the Gleason–Yamabe theorem and then uses a closed normal subgroup of type I with abelian quotient, constructed by Dixmier.

These results are used in the lessons "Uniqueness of the injective II₁ factor" and "Classification of injective factors" of this course, where injectivity is shown to coincide with approximate finite dimensionality for factors with separable predual.

What is assumed. The lesson uses completely positive maps, the double commutant theorem, the basic theory of normal representations, direct integrals and amenable groups. These facts are listed in "Results used from other lessons", each with the lesson that proves it. Section 7 also uses three results from the structure theory of locally compact groups and Lie groups, (L1)–(L3). It proves (L1) from the course on KK-theory and quotes (L2) and (L3). The lesson does not depend on the earlier lessons of this course.

Basic references are [Connes 1976], [Connes 1994], [Arveson 1969], [Hakeda–Tomiyama 1967] and [Anantharaman–Popa].

Conventions. All C*-algebras have a unit, and a C*-subalgebra \(B\subset C\) contains the unit of \(C\). A von Neumann algebra in \(H\) contains \(1_H\). For a C*-algebra \(A\) we write \(A_{sa}\) for its self-adjoint part and \(M_n(A)\) for the \(n\times n\) matrices over \(A\); \(M_n=M_n(\mathbb C)\). For \(x\in B(H)\) and \(\sigma=(\sigma_{ij})\in M_n\) we write \[ x\otimes\sigma=(\sigma_{ij}x)_{i,j}\in M_n(B(H))=B(H\otimes\mathbb C^n). \] For a linear map \(\varphi\) we write \(\varphi_n=\varphi\otimes\mathrm{id}_{M_n}\), the map applied entrywise to matrices. Note that \(\varphi_n(x\otimes\sigma)=\varphi(x)\otimes\sigma\) and that \(\varphi_n\) is the identity on \(M_n(B)\) when \(\varphi\) is the identity on \(B\).

Results used from other lessons

(B1) Completely positive maps. An operator system is a self-adjoint linear subspace \(V\) of a C*-algebra \(C\) containing the unit (it need not be closed). A linear map \(\varphi\colon V\to B\) into a C*-algebra is completely positive if every \(\varphi_n\colon M_n(V)\to M_n(B)\) maps positive elements (positivity taken in \(M_n(C)\)) to positive elements. A positive map on an operator system satisfies \(\varphi(v^*)=\varphi(v)^*\). A completely positive map \(\varphi\) on a C*-algebra is bounded with \(\|\varphi\|=\|\varphi(1)\|\); in particular a unital completely positive map is contractive. Compositions of completely positive maps are completely positive. We write "ucp" for "unital completely positive". For maps between C*-algebras these facts are proved in Completely positive maps, Proposition 3.2 and Theorem 4.1(2). For a positive map on an operator system, a self-adjoint \(v\in V\) is the difference of the positive elements \((\|v\|1\pm v)/2\) of \(V\), so \(\varphi(v)\) is self-adjoint, and a general \(v\) is \(h+ik\) with \(h,k\) self-adjoint elements of \(V\); this gives \(\varphi(v^*)=\varphi(v)^*\).

(B2) Arveson's extension theorem. Let \(V\) be an operator system in a C*-algebra \(C\), and \(\varphi\colon V\to B(K)\) completely positive. Then there is a completely positive \(\Phi\colon C\to B(K)\) extending \(\varphi\). If \(\varphi\) is unital, so is \(\Phi\). Proved, for every Hilbert space \(K\), in Completely positive finite models, Theorem 2.3. See also [Arveson 1969].

(B3) Tomiyama's theorem on projections of norm one. Let \(B\subset C\) be C*-algebras and \(E\colon C\to B\) a linear map with \(E(b)=b\) for \(b\in B\) and \(\|E\|\le1\). Then \(E\) is completely positive and unital, and \(E(b_1cb_2)=b_1E(c)b_2\) for \(b_1,b_2\in B\), \(c\in C\). Positivity, the bimodule property and \(E(1)=1\) are proved in The universal enveloping von Neumann algebra of a C*-algebra, and W*-algebras, Theorem 8.5. Complete positivity follows from them: for \(x_1,\dots,x_n\in C\) and \(y_1,\dots,y_n\in B\), \(\sum_{i,j}y_i^*E(x_i^*x_j)y_j=E\bigl((\sum_jx_jy_j)^*(\sum_jx_jy_j)\bigr)\ge0\), which is the criterion of Completely positive maps, Proposition 3.2(1). It is also proved in Contractive retractions and the algebraic structure of expectations. See also [Tomiyama 1957].

(B4) The σ-weak topology. On \(B(H)\) the σ-weakly continuous linear functionals are \(x\mapsto\operatorname{Tr}(\rho x)\) with \(\rho\) of trace class; they are positive exactly when \(\rho\ge0\). On bounded sets the σ-weak and weak operator topologies agree. The unit ball of a von Neumann algebra is σ-weakly compact, and the positive cone of \(B(H)\) is σ-weakly closed. In a locally convex space, a compact convex set and a disjoint closed convex set are strictly separated by a continuous real linear functional (the Hahn–Banach separation theorem). Proved in Compact and trace-class operators, the predual of B(H), and the operator topologies: the σ-weakly continuous functionals are the normal ones \(x\mapsto\operatorname{Tr}(\rho x)\) (Definition 6.1 and Proposition 6.3(a), Theorem 9.1), the two topologies agree on bounded sets (Lemma 8.5), and the unit ball of \(B(H)\) is σ-weakly compact (Corollary 9.7(a)), so the unit ball of a von Neumann algebra, a σ-weakly closed subset of it, is σ-weakly compact. The positive cone is σ-weakly closed because each \(x\mapsto\langle x\xi,\xi\rangle\) is σ-weakly continuous. The separation theorem is Hahn–Banach, Baire and the basic theorems on Banach spaces, Theorem 6.3.

(B5) Normal representations. The image of a von Neumann algebra under a faithful normal \(\ast\)-representation is a von Neumann algebra. If \(\pi_1\colon M\to B(H_1)\) and \(\pi_2\colon M\to B(H_2)\) are faithful normal representations, there are a Hilbert space \(K\), a projection \(e'\) in the commutant of \(\pi_1(M)\otimes1_K\) and a unitary \(U\colon e'(H_1\otimes K)\to H_2\) with \(\pi_2(x)=U(\pi_1(x)\otimes1_K)e'U^*\) for all \(x\in M\). For a von Neumann algebra \(N\subset B(L)\) and a projection \(e'\in N'\), the reduced algebra \(N_{e'}=Ne'\) on \(e'L\) has commutant \((N_{e'})'=e'N'e'\). Finally \((1_L\otimes B(K))'=B(L)\otimes1_K\). These facts are proved in Spatial tensor products of von Neumann algebras: the second statement is Theorem 8.2 applied to the normal \(\ast\)-isomorphism \(\pi_2\circ\pi_1^{-1}\) of the von Neumann algebra \(\pi_1(M)\) (it gives an isometry \(V\colon H_2\to H_1\otimes K\) with \(e'=VV^*\), and \(U=V^*\) on \(e'(H_1\otimes K)\)), and it also gives the first statement; the reduced algebra is Proposition 6.1(2); the last identity is Proposition 7.1 with \(M=\mathbb C1_L\). The inverse \(\pi_1^{-1}\) is normal because every \(\ast\)-isomorphism of von Neumann algebras is (The universal enveloping von Neumann algebra of a C*-algebra, and W*-algebras, Corollary 11.4).

(B6) Standard form. Every von Neumann algebra \(M\) has a faithful normal representation \(\pi\) on a Hilbert space \(L\) together with a conjugate-linear isometry \(J\) of \(L\) onto itself with \(J^2=1\) and \(J\pi(M)J=\pi(M)'\). This is part of Tomita–Takesaki theory, proved in the course Modular theory and weights: every von Neumann algebra has a faithful normal semifinite weight (Weights and the Hilbert spaces of multiplication, §13); its GNS representation is faithful and normal and carries a full left Hilbert algebra (§8–§11); and the modular conjugation \(J\) of that algebra satisfies \(J^2=1\) and \(JMJ=M'\) (The modular group and its analytic algebra, §5).

(B7) Type I algebras. Every von Neumann algebra of type I is isomorphic to a product \(\prod_{j}A_j\bar\otimes B(K_j)\) with \(A_j\) abelian von Neumann algebras. A finite-dimensional C*-algebra is a finite product of full matrix algebras \(M_{n}\). The first statement is proved in Projections and types of von Neumann algebras, Theorem 10.3, the second in AF-algebras, Section 2.

(B8) Amenable groups. A locally compact group \(G\) is amenable if there is a state \(m\) on \(L^\infty(G)\) (a mean) with \(m(f(h\,\cdot\,))=m(f)\) for all \(f\) and \(h\in G\). Since Haar measure charges every nonempty open set, the bounded continuous functions \(C_b(G)\) embed in \(L^\infty(G)\), compatibly with translations, so an amenable group has a left-invariant mean on \(C_b(G)\). For a discrete group, \(L^\infty(G)=\ell^\infty(G)\), and "amenable as a discrete group" refers to this case. Abelian groups and compact groups are amenable; subgroups of amenable discrete groups are amenable; the free group \(\mathbb F_2\) on two generators is not amenable; the rotation group \(SO(3)\) contains a copy of \(\mathbb F_2\), so it is amenable (being compact) but not amenable as a discrete group. Haar measure charges every nonempty open set by Haar measure on locally compact groups, Proposition 9.1. Abelian groups are amenable by Amenability and the equality of full and reduced crossed products: every locally compact abelian group has almost invariant unit vectors in \(L^2(G)\), and these give an invariant mean on \(L^\infty(G)\) (Definition 4.1 and the argument after it). A compact group is amenable because integration against normalized Haar measure is an invariant mean. A subgroup \(\Lambda\) of an amenable discrete group \(\Gamma\) is amenable: choose a set \(R\) of representatives of the right cosets \(\Lambda\backslash\Gamma\), so that every \(g\in\Gamma\) is \(\lambda(g)r(g)\) with \(\lambda(g)\in\Lambda\), \(r(g)\in R\), and put \(m_\Lambda(f)=m(f\circ\lambda)\); since \(\lambda(hg)=h\lambda(g)\) for \(h\in\Lambda\), this is a left-invariant mean on \(\ell^\infty(\Lambda)\). The statements on \(\mathbb F_2\) and \(SO(3)\) are used only in examples: \(\mathbb F_2\) is not amenable ([Bekka–de la Harpe–Valette, Example G.2.4(ii)]), and \(SO(3)\) contains a free subgroup on two generators (a classical observation of Hausdorff), so by the subgroup statement above \(SO(3)\), as a discrete group, is not amenable.

(B9) Borel structure on the unit ball. Let \(H\) be separable and \(B_r=\{x\in B(H):\|x\|\le r\}\). With the strong* topology \(B_r\) is a Polish space; a compatible metric is \(d(x,y)=\sum_j2^{-j}(\|(x-y)\xi_j\|+\|(x-y)^*\xi_j\|)\) for a dense sequence \((\xi_j)\) in the unit ball of \(H\). With the weak operator topology \(B_r\) is compact metrizable, and the two topologies have the same Borel sets. On \(B_r\) multiplication is jointly strong* continuous and the adjoint is strong* continuous. A map \(f\) from a Borel space into \(B_r\) is Borel exactly when \(\alpha\mapsto\langle f(\alpha)\xi,\eta\rangle\) is Borel for all \(\xi,\eta\in H\). The positive trace-class operators \(\rho\) with \(\operatorname{Tr}\rho\le k\), with the trace norm, form a Polish space. The metrics and the continuity of the operations are in Compact and trace-class operators, the predual of B(H), and the operator topologies, Propositions 10.1(a) and 8.6, and weak compactness of \(B_r\) is Corollary 9.7(a). The metric \(d\) is complete on \(B_r\): if \((x_n)\) is \(d\)-Cauchy, then \((x_n\xi)\) and \((x_n^*\xi)\) converge for each \(\xi_j\), hence, the \(x_n\) being bounded by \(r\), for every \(\xi\in H\); the limits define \(x\in B_r\) and \(x^*\), and \(x_n\to x\) strongly*. It is separable, since the compressions \(p_kxp_k\) to the spans of the first \(k\) vectors of an orthonormal basis converge strongly* to \(x\), and they are limits of matrices with rational entries in the ball. The identity map from \(B_r\) with the strong* topology onto \(B_r\) with the weak topology is a continuous bijection of Polish spaces, so it carries Borel sets to Borel sets (Polish spaces and standard Borel spaces, Theorem 3.6): the two topologies have the same Borel sets. With the weak topology, \(B_r\) is homeomorphic, through \(x\mapsto(\langle x\xi_j,\xi_k\rangle)_{j,k}\), to a subset of a countable product of discs, so its Borel sets are generated by these coefficients, and by density by all \(x\mapsto\langle x\xi,\eta\rangle\). Finally the trace-class operators form a separable Banach space when \(H\) is separable, since finite sums of vector functionals are dense (Theorem 6.2), and the positive ones of trace at most \(k\) form a closed subset.

(B10) Direct integrals. Let \(H\) be separable and \((X,\mu)\) a standard Borel space with a probability measure. A bounded Borel map \(x\colon X\to B(H)\) (a bounded Borel field) defines a bounded operator on \(L^2(X,\mu;H)\) by \((x\xi)(\alpha)=x(\alpha)\xi(\alpha)\), of norm the essential supremum of \(\|x(\alpha)\|\); two fields define the same operator exactly when they agree almost everywhere, and the assignment respects sums, products and adjoints. The operators so obtained, the decomposable operators, form the von Neumann algebra \(D=(L^\infty(X,\mu)\otimes1_H)'\). Identifying \(M_n(D)\) with the decomposable operators on \(L^2(X,\mu;H\otimes\mathbb C^n)\), the field of \((x_{ij})\) is \(\alpha\mapsto(x_{ij}(\alpha))\), and an element of \(M_n(D)\) is positive exactly when its field is positive almost everywhere. A map \(\alpha\mapsto M(\alpha)\) from \(X\) to von Neumann algebras in \(H\) is Borel for the Effros Borel structure exactly when there are bounded Borel fields \(a_1,a_2,\dots\colon X\to B_1\) such that for every \(\alpha\) the set \(\{a_k(\alpha)\}_k\) is strong* dense in the unit ball of \(M(\alpha)\). The set of \(x\in D\) whose field satisfies \(x(\alpha)\in M(\alpha)\) for almost every \(\alpha\) is then a von Neumann algebra, the direct integral \(\int^\oplus_XM(\alpha)\,d\mu(\alpha)\); it contains \(L^\infty(X,\mu)\otimes1_H\). These facts are proved in the course Measurable fields and direct integrals: the operator of a bounded field, its norm, and the rules for sums, products and adjoints in Measurable fields of Hilbert spaces and their direct integrals, Theorem 10.1; decomposability as commuting with \(L^\infty(X,\mu)\otimes1_H\) in Decomposable operators and the diagonal algebra, Theorem 5.1; the matrix fields by Measurable fields of Hilbert spaces and their direct integrals, Proposition 7.1(3); the Borel criterion in The Effros Borel structure, Theorem 10.3(1)–(2), which gives fields with \(\{a_k(\alpha)\}\) σ-weakly dense in the unit ball, and rational convex combinations of these fields are then strong* dense, because a convex set has the same σ-weak and strong* closures (Compact and trace-class operators, Theorem 9.6(a)); and the direct integral in Direct integrals of von Neumann algebras, Proposition 2.2(2) and Theorem 3.2(2). A decomposable operator is positive exactly when its field is positive almost everywhere: if \(x(\alpha)\ge0\) almost everywhere, then \(\langle x\xi,\xi\rangle=\int\langle x(\alpha)\xi(\alpha),\xi(\alpha)\rangle\,d\mu\ge0\); conversely, if \(x\ge0\), then for each Borel set \(E\) and each \(v\) in a countable dense subset \(D\) of \(H\otimes\mathbb C^n\), \(\int_E\langle x(\alpha)v,v\rangle\,d\mu=\langle x(1_E\otimes v),1_E\otimes v\rangle\ge0\), so \(\langle x(\alpha)v,v\rangle\ge0\) for almost every \(\alpha\), simultaneously for all \(v\in D\), and \(x(\alpha)\ge0\) by density. See also [Effros 1965].

(B11) Measurable selection. Let \(X\) and \(Y\) be standard Borel spaces, \(\mu\) a probability measure on \(X\), and \(G\subset X\times Y\) a Borel set. Then the projection \(p(G)=\{\alpha:\exists y,\ (\alpha,y)\in G\}\) is an analytic set; analytic sets are \(\mu\)-measurable, and countable unions of them are analytic. There are a Borel set \(X'\subset p(G)\) with \(\mu(p(G)\setminus X')=0\) and a Borel map \(f\colon X'\to Y\) with \((\alpha,f(\alpha))\in G\) for all \(\alpha\in X'\). This is the Jankov–von Neumann uniformization theorem. It is proved in Polish spaces and standard Borel spaces: \(p(G)\) is the image of the standard Borel space \(G\) under a Borel map, hence a Souslin (analytic) set, and there is a section \(\varphi\colon p(G)\to G\) whose preimages of Borel sets are \(\mu\)-measurable (Theorem 7.7); analytic sets are \(\mu\)-measurable (Theorem 6.2) and countable unions of analytic sets are analytic (Proposition 2.2). The second coordinate of \(\varphi\) is a \(\mu\)-measurable map into the standard Borel space \(Y\); it agrees with a Borel map on a Borel set \(X'\subset p(G)\) of full measure, which gives \(f\).

1. Injective C*-algebras

A reference for this section is [Arveson 1969].

Definition 1.1. A C*-algebra \(A\) is injective if for all C*-algebras \(B\subset C\) and every ucp map \(\theta\colon B\to A\) there is a ucp map \(\bar\theta\colon C\to A\) with \(\bar\theta|_B=\theta\).

Definition 1.2. We call a von Neumann algebra injective if it is injective as a C*-algebra.

The definition refers only to the C*-algebra structure. So if two von Neumann algebras are \(\ast\)-isomorphic, one is injective exactly when the other is; injectivity does not depend on the Hilbert space in which the algebra acts.

Example 1.3.

(a) \(B(K)\) is injective for every Hilbert space \(K\). This is Arveson's extension theorem (B2) applied to \(V=B\).

(b) In particular \(\mathbb C=B(\mathbb C)\) and \(M_n=B(\mathbb C^n)\) are injective. For \(\mathbb C\) one can see this directly: a ucp map into \(\mathbb C\) is a state, and states extend to states by the Hahn–Banach theorem together with the fact that a functional \(\omega\) on a C*-algebra with \(\omega(1)=\|\omega\|\) is positive.

Proposition 1.4. A C*-algebra \(A\) is injective if and only if its opposite algebra \(A^{op}\) is injective.

Proof. Let \(A\) be any C*-algebra. The positive elements of \(A^{op}\) and of \(A\) are the same: the involution is the same and \(x^*\circ x=xx^*\), where \(\circ\) is the product of \(A^{op}\). The transpose \(t\colon M_n(A^{op})\to M_n(A)^{op}\), \(t((a_{ij}))=(a_{ji})\), is a \(\ast\)-isomorphism: in \(M_n(A^{op})\) the \((j,i)\) entry of \(a\circ b\) is \(\sum_kb_{ki}a_{jk}\), which is the \((i,j)\) entry of \(t(b)t(a)\), that is of \(t(a)\circ t(b)\); and \(t(a^*)_{ij}=a_{ij}^*=(t(a)^*)_{ij}\). Hence \(a\in M_n(A^{op})\) is positive exactly when \(t(a)\) is positive in \(M_n(A)\). A linear map \(\varphi\) applied entrywise commutes with \(t\). It follows that a linear map \(\varphi\colon A_1\to A_2\) is completely positive exactly when the same map \(\varphi\colon A_1^{op}\to A_2^{op}\) is.

Now let \(A\) be injective, \(B\subset C\), and \(\theta\colon B\to A^{op}\) ucp. Then \(\theta\colon B^{op}\to A\) is ucp, it extends to a ucp map \(C^{op}\to A\), and this is a ucp map \(C\to A^{op}\) extending \(\theta\). The converse follows since \((A^{op})^{op}=A\). \(\square\)

2. Injectivity as a norm-one projection

References for this section are [Tomiyama 1957], [Hakeda–Tomiyama 1967] and [Connes 1976, §4].

Theorem 2.1. A von Neumann algebra \(M\subset B(H)\) is injective if and only if there is a linear map \(E\colon B(H)\to M\) of norm one with \(E(x)=x\) for \(x\in M\). Every such \(E\) is completely positive, unital and satisfies \(E(axb)=aE(x)b\) for \(a,b\in M\), \(x\in B(H)\).

Proof. Suppose \(M\) is injective. Apply Definition 1.1 with \(B=M\subset C=B(H)\) and \(\theta=\mathrm{id}_M\). This gives a ucp map \(E\colon B(H)\to M\) that is the identity on \(M\); by (B1) \(\|E\|=\|E(1)\|=1\).

Conversely, let \(E\) be a projection of norm one onto \(M\). By Tomiyama's theorem (B3) \(E\) is completely positive, unital and \(M\)-bimodular; this proves the last sentence. Let \(B\subset C\) be C*-algebras and \(\theta\colon B\to M\) ucp. Regard \(\theta\) as a map into \(B(H)\). By Arveson's extension theorem (B2) there is a ucp map \(\Theta\colon C\to B(H)\) extending \(\theta\). Then \(E\circ\Theta\colon C\to M\) is ucp, and for \(b\in B\) we have \(E(\Theta(b))=E(\theta(b))=\theta(b)\) since \(\theta(b)\in M\). \(\square\)

Corollary 2.2.

(a) Whether \(B(H)\) admits a projection of norm one onto a von Neumann algebra \(M\subset B(H)\) depends only on the isomorphism class of \(M\), not on \(H\).

(b) An injective von Neumann algebra \(M\) has the extension property for operator systems: if \(V\subset C\) is an operator system in a C*-algebra and \(\theta\colon V\to M\) is ucp, then \(\theta\) extends to a ucp map \(C\to M\).

Proof. (a) is Theorem 2.1 together with the remark after Definition 1.2. For (b), represent \(M\subset B(H)\), let \(E\) be as in Theorem 2.1, extend \(\theta\) to a ucp map \(\Theta\colon C\to B(H)\) by (B2), and take \(E\circ\Theta\). \(\square\)

Statement (a) is not obvious from the projection property itself: two representations of \(M\) can look very different (for instance \(M\) and \(M\otimes1_K\)), and nothing in the definition of a projection relates them.

Proposition 2.3. Let \(N\subset B(L)\) be a von Neumann algebra.

(a) If \(N\) is injective and \(e\in N\) is a nonzero projection, then \(eNe\), acting on \(eL\), is injective.

(b) A product \(\prod_jN_j\) of von Neumann algebras is injective if and only if every \(N_j\) is injective.

(c) If \(N\) is injective and \(K\) is any Hilbert space, then \(N\bar\otimes B(K)\) is injective. Here \(N\bar\otimes B(K)\) is the von Neumann algebra on \(L\otimes K\) generated by \(N\otimes1_K\) and \(1_L\otimes B(K)\); it equals \((N'\otimes1_K)'\).

(d) Conversely, if \(N\bar\otimes B(K)\) is injective for some \(K\ne0\), then \(N\) is injective.

Proof. (a) Let \(E\colon B(L)\to N\) be as in Theorem 2.1. For \(y\in B(eL)=eB(L)e\), bimodularity gives \(E(y)=E(eye)=eE(y)e\in eNe\). So \(E\) restricts to a ucp map \(B(eL)\to eNe\) (it is unital because \(E(e)=e\), and \(e\) is the unit of \(B(eL)\)), which is the identity on \(eNe\). By Theorem 2.1, \(eNe\) is injective.

(b) Represent each \(N_j\) on \(H_j\) with a projection of norm one \(E_j\colon B(H_j)\to N_j\) (Theorem 2.1), and let \(p_j\) be the projection of \(\bigoplus_jH_j\) onto \(H_j\). Put \(E(x)=\bigoplus_jE_j(p_jxp_j)\) for \(x\in B(\bigoplus_jH_j)\); the sum is bounded since \(\|E_j(p_jxp_j)\|\le\|x\|\). If \(x\in M_n(B(\bigoplus H_j))\) is positive, each compression \((p_jx_{kl}p_j)_{k,l}\) is positive, hence so is each \(\bigl(E_j(p_jx_{kl}p_j)\bigr)_{k,l}\) and their direct sum. So \(E\) is ucp with values in \(\prod_jN_j\), and it is the identity on \(\prod_jN_j\). For the converse, \(N_j\) is isomorphic to the corner of \(\prod_iN_i\) by the central projection \(p_j\); apply (a).

(c) First the commutant. By (B5), \((1_L\otimes B(K))'=B(L)\otimes1_K\), and the elements of \(B(L)\otimes1_K\) that commute with \(N\otimes1_K\) are those of \(N'\otimes1_K\). Hence the commutant of \(N\bar\otimes B(K)\) is \(N'\otimes1_K\), and \(N\bar\otimes B(K)=(N'\otimes1_K)'\) by the double commutant theorem. Fix an orthonormal basis \((e_j)_{j\in J}\) of \(K\) and let \(V_j\colon L\to L\otimes K\), \(V_j\xi=\xi\otimes e_j\). For \(x\in B(L\otimes K)\) put \(x_{ij}=V_i^*xV_j\in B(L)\). Since \((a\otimes1)V_j=V_ja\), an operator \(x\) commutes with \(N'\otimes1\) exactly when every \(x_{ij}\) commutes with \(N'\). Thus \[ N\bar\otimes B(K)=\{x\in B(L\otimes K):x_{ij}\in N\ \text{for all }i,j\}. \tag{2.1} \] Let \(E\colon B(L)\to N\) be a projection of norm one. For finite \(F\subset J\), the map \(x\mapsto(x_{ij})_{i,j\in F}\) identifies \(P_FB(L\otimes K)P_F\), where \(P_F=\sum_{j\in F}V_jV_j^*\), with \(M_F(B(L))\), and \(E_F:=E\otimes\mathrm{id}\) is ucp on it. Put \(y_F=\sum_{i,j\in F}V_iE(x_{ij})V_j^*\). Then \(\|y_F\|\le\|P_FxP_F\|\le\|x\|\), and \(y_F\ge0\) when \(x\ge0\). For vectors \(\xi,\eta\) with finitely many nonzero coordinates, \(\langle y_F\xi,\eta\rangle\) is constant once \(F\) contains their supports. A bounded net that converges on a dense set of vectors converges in the weak operator topology, so \(\tilde E(x):=\lim_Fy_F\) exists, with \(\|\tilde E(x)\|\le\|x\|\), \(\tilde E(x)_{ij}=E(x_{ij})\), and \(\tilde E(x)\ge0\) when \(x\ge0\). By (2.1), \(\tilde E(x)\in N\bar\otimes B(K)\), and \(\tilde E(x)=x\) when all \(x_{ij}\in N\). \(\tilde E(1)=1\) since \(1_{ij}=\delta_{ij}1\). Finally \(M_n(B(L\otimes K))=B(L\otimes K\otimes\mathbb C^n)\), and \(\tilde E_n\) is the same construction for the Hilbert space \(K\otimes\mathbb C^n\) with the basis \((e_j\otimes f_k)\), where \((f_k)\) is the standard basis of \(\mathbb C^n\), so \(\tilde E_n\) is positive. Thus \(\tilde E\) is a ucp projection onto \(N\bar\otimes B(K)\).

(d) Let \(p\) be a rank-one projection in \(B(K)\). Then \(e=1\otimes p\) lies in \(N\bar\otimes B(K)\), and \(e(N\bar\otimes B(K))e=N\otimes p\) is isomorphic to \(N\). Apply (a). \(\square\)

Example 2.4. Every finite-dimensional C*-algebra is injective: by (B7) it is a finite product of algebras \(M_n=\mathbb C\bar\otimes B(\mathbb C^n)\), and Example 1.3 and Proposition 2.3(b) apply. Also \(\mathbb C1_H\subset B(H)\) is the range of the ucp projection \(x\mapsto\omega(x)1_H\) for any state \(\omega\).

3. A criterion by matrix inequalities

In this section \(M\subset B(H)\) is a von Neumann algebra. The criterion is due to Choi and Effros; it is stated in [Connes 1976, §4].

Definition 3.1. Let \(b\in B(H)_{sa}\). A \(b\)-constraint is a triple \((n,s,\sigma)\) with \(n\ge1\), \(s\in M_n(M)_{sa}\), \(\sigma\in(M_n)_{sa}\) and \[ b\otimes\sigma\le s. \tag{3.1} \] An element \(x\in M_{sa}\) solves it if \(\|x\|\le\|b\|\) and \(x\otimes\sigma\le s\).

The constraint (3.1) is an inequality between operators on \(H\otimes\mathbb C^n\) that involves the operator \(b\), which is not in \(M\). A solution replaces \(b\) by an element of \(M\) of no larger norm.

Lemma 3.2. Let \(b\in B(H)_{sa}\) and suppose that every \(b\)-constraint has a solution. Then there is a single \(x\in M_{sa}\) with \(\|x\|\le\|b\|\) that solves every \(b\)-constraint.

Proof. For a \(b\)-constraint \(c=(n,s,\sigma)\) let \(S_c\) be its set of solutions. It is nonempty by hypothesis. It is σ-weakly closed: the map \(x\mapsto s-x\otimes\sigma\) from \(M\) to \(M_n(B(H))\) is σ-weakly continuous, and the positive cone and the self-adjoint part are σ-weakly closed. It lies in the ball of radius \(\|b\|\) of \(M\), which is σ-weakly compact (B4). So it suffices to show that finitely many of the sets \(S_c\) always intersect.

Let \(c_k=(n_k,s_k,\sigma_k)\), \(k=1,\dots,m\), be \(b\)-constraints. Put \(n=n_1+\dots+n_m\) and form the block diagonal matrices \(s=s_1\oplus\dots\oplus s_m\in M_n(M)_{sa}\) and \(\sigma=\sigma_1\oplus\dots\oplus\sigma_m\in (M_n)_{sa}\). Then \(b\otimes\sigma=(b\otimes\sigma_1)\oplus\dots\oplus(b\otimes\sigma_m)\le s\), so \((n,s,\sigma)\) is a \(b\)-constraint. A solution \(x\) of it satisfies \((x\otimes\sigma_1)\oplus\dots\oplus(x\otimes\sigma_m)\le s_1\oplus\dots\oplus s_m\), that is, \(x\otimes\sigma_k\le s_k\) for every \(k\). So \(x\in S_{c_1}\cap\dots\cap S_{c_m}\). \(\square\)

Lemma 3.3 (one-step extension). Let \(b\in B(H)_{sa}\) with \(b\notin M\), and let \(x\in M_{sa}\). The map \[ \psi\colon M+\mathbb Cb\to M,\qquad \psi(m+\lambda b)=m+\lambda x\quad(m\in M,\ \lambda\in\mathbb C), \] is completely positive if and only if \(x\otimes\sigma\le s\) for every \(b\)-constraint \((n,s,\sigma)\). In that case \(\|x\|\le\|b\|\).

Proof. The sum \(M+\mathbb Cb\) is direct, so \(\psi\) is well defined; it is an operator system and \(\psi\) is unital. Every element of \(M_n(M+\mathbb Cb)\) can be written uniquely as \(w=m+b\otimes\Lambda\) with \(m\in M_n(M)\) and \(\Lambda\in M_n\), and then \(\psi_n(w)=m+x\otimes\Lambda\). If \(w\) is self-adjoint, comparing \(w\) with \(w^*=m^*+b\otimes\Lambda^*\) entrywise gives \(m_{ij}-(m^*)_{ij}=(\overline{\Lambda_{ji}}-\Lambda_{ij})b\); the left side lies in \(M\) and \(b\notin M\), so \(\Lambda=\Lambda^*\) and \(m=m^*\). For self-adjoint \(w\), \[ w\ge0\iff b\otimes(-\Lambda)\le m,\qquad \psi_n(w)\ge0\iff x\otimes(-\Lambda)\le m. \] Positive elements are self-adjoint, so \(\psi\) is completely positive exactly when every \(b\)-constraint \((n,m,-\Lambda)\) satisfies \(x\otimes(-\Lambda)\le m\); and every \(b\)-constraint has this form. Finally \(-\|b\|\le b\le\|b\|\) says that \((1,\|b\|,1)\) and \((1,\|b\|,-1)\) are \(b\)-constraints, so \(-\|b\|\le x\le\|b\|\), that is, \(\|x\|\le\|b\|\). \(\square\)

Theorem 3.4 (the matrix inequality criterion). For a von Neumann algebra \(M\subset B(H)\) the following are equivalent.

(i) \(M\) is injective.

(ii) For every \(b\in B(H)_{sa}\), every \(b\)-constraint has a solution. Explicitly: whenever \(n\ge1\), \(s\in M_n(M)_{sa}\), \(\sigma\in(M_n)_{sa}\) and \(b\otimes\sigma\le s\) for some \(b\in B(H)_{sa}\), there is \(x\in M_{sa}\) with \(x\otimes\sigma\le s\) and \(\|x\|\le\|b\|\).

(iii) For every \(b\in B(H)_{sa}\) there is one \(x\in M_{sa}\) with \(\|x\|\le\|b\|\) that solves all \(b\)-constraints.

Reference: [Connes 1976, §4], where it is credited to Choi and Effros.

Proof. (i)⇒(ii). Let \(E\colon B(H)\to M\) be a projection of norm one; it is ucp and the identity on \(M\) (Theorem 2.1). If \(b\otimes\sigma\le s\), apply the positive map \(E_n\) to \(s-b\otimes\sigma\ge0\): \(E_n(s)-E(b)\otimes\sigma=s-E(b)\otimes\sigma\ge0\). So \(x=E(b)\) solves the constraint: it is self-adjoint since \(E\) is positive, and \(\|E(b)\|\le\|b\|\).

(ii)⇒(iii) is Lemma 3.2.

(iii)⇒(i). Consider the pairs \((V,\varphi)\) where \(V\) is an operator system with \(M\subset V\subset B(H)\) and \(\varphi\colon V\to M\) is ucp with \(\varphi|_M=\mathrm{id}_M\). Order them by extension. The pair \((M,\mathrm{id}_M)\) is one of them. If \((V_i,\varphi_i)\) is a chain, let \(V=\bigcup_iV_i\) and let \(\varphi\) be the common extension. It is ucp: a positive element of \(M_n(V)\) has finitely many entries, so it lies in some \(M_n(V_i)\), and \(\varphi_n\) agrees there with the positive map \((\varphi_i)_n\). By Zorn's lemma there is a maximal pair \((V,\varphi)\).

Suppose \(V\ne B(H)\). Since \(B(H)\) is spanned by its self-adjoint elements and \(V\) is self-adjoint, there is \(b\in B(H)_{sa}\) with \(b\notin V\). Regard \(\varphi\) as a ucp map into \(B(H)\) and extend it, by (B2), to a ucp map \(\Phi\colon B(H)\to B(H)\). Put \(b'=\Phi(b)\in B(H)_{sa}\). If \(b'\in M\) put \(x=b'\); otherwise take \(x\in M_{sa}\) as in (iii) for \(b'\), so that by Lemma 3.3 the map \(\psi\colon M+\mathbb Cb'\to M\), \(m+\lambda b'\mapsto m+\lambda x\), is ucp. In both cases define \[ \tilde\varphi\colon V+\mathbb Cb\to M,\qquad\tilde\varphi(v+\lambda b)=\varphi(v)+\lambda x . \] It is well defined since \(b\notin V\), unital, and extends \(\varphi\). It is completely positive: \(\Phi\) maps \(V+\mathbb Cb\) into \(M+\mathbb Cb'\), and \(\tilde\varphi=\psi\circ\Phi\) on \(V+\mathbb Cb\) (or \(\tilde\varphi=\Phi\) when \(b'\in M\)), a composition of completely positive maps. This contradicts maximality. Hence \(V=B(H)\), and \(\varphi\colon B(H)\to M\) is a ucp map that is the identity on \(M\); its norm is \(\|\varphi(1)\|=1\). By Theorem 2.1, \(M\) is injective. \(\square\)

Example 3.5 (the scalar level carries no information). For \(n=1\), every \(b\)-constraint has a solution, for any von Neumann algebra \(M\). Indeed, let \(\sigma\in\mathbb R\) and \(\sigma b\le s\). If \(\sigma>0\), put \(x=-\|b\|\): since \(b\ge-\|b\|\), \(x\sigma\le\sigma b\le s\). If \(\sigma<0\), put \(x=\|b\|\): since \(b\le\|b\|\), \(x\sigma\le\sigma b\le s\). If \(\sigma=0\), put \(x=0\). So Theorem 3.4 genuinely needs matrices: positivity alone is not enough, complete positivity is what is being tested. By Section 5, the von Neumann algebra of the free group on two generators satisfies the scalar condition but is not injective.

4. Commutants and monotone limits

References for this section are [Hakeda–Tomiyama 1967] and [Connes 1976, §4].

Lemma 4.1. Let \(\pi_1,\pi_2\) be faithful normal representations of a von Neumann algebra \(M\). If \(\pi_1(M)'\) is injective, so is \(\pi_2(M)'\).

Proof. Take \(K\), \(e'\) and \(U\) as in (B5). By Proposition 2.3(c), applied to \(N=\pi_1(M)'\) (so that \(N'=\pi_1(M)\)), the algebra \(Q:=(\pi_1(M)\otimes1_K)'=\pi_1(M)'\bar\otimes B(K)\) is injective. The projection \(e'\) lies in \(Q\). The reduced algebra \((\pi_1(M)\otimes1_K)_{e'}\) has commutant \(e'Qe'\) by (B5), and \(e'Qe'\) is injective by Proposition 2.3(a). Conjugation by \(U\) carries \((\pi_1(M)\otimes1_K)_{e'}\) onto \(\pi_2(M)\), hence carries \(e'Qe'\) onto \(\pi_2(M)'\). Injectivity is invariant under isomorphism, so \(\pi_2(M)'\) is injective. \(\square\)

Theorem 4.2. A von Neumann algebra \(M\subset B(H)\) is injective if and only if its commutant \(M'\) is injective.

Reference: Hakeda and Tomiyama proved the semifinite case [Hakeda–Tomiyama 1967]. The proof below covers every von Neumann algebra, type III included: in a standard form the commutant is a copy of \(M^{op}\), and Lemma 4.1 moves injectivity of the commutant between representations.

Proof. Let \(M\) be injective, and let \((\pi,L,J)\) be a standard form of \(M\) (B6). The map \(x\mapsto J\pi(x)^*J\) is linear (as \(J\) is conjugate-linear), preserves adjoints, and reverses products: \(J\pi(xy)^*J=J\pi(y)^*JJ\pi(x)^*J\). It is injective and its image is \(J\pi(M)J=\pi(M)'\). So \(\pi(M)'\) is isomorphic to \(M^{op}\), which is injective by Proposition 1.4. By Lemma 4.1, applied with \(\pi_1=\pi\) and \(\pi_2\) the identity representation of \(M\) on \(H\), the commutant \(M'\) is injective. Conversely, if \(M'\) is injective, then so is \(M''=M\). \(\square\)

Theorem 4.3. Let \((M_i)_{i\in I}\) be a family of von Neumann algebras in \(H\), indexed by a directed set.

(a) If the family is decreasing (\(M_j\subset M_i\) for \(i\le j\)) and every \(M_i\) is injective, then \(\bigcap_iM_i\) is injective.

(b) If the family is increasing (\(M_i\subset M_j\) for \(i\le j\)) and every \(M_i\) is injective, then the von Neumann algebra \(M=\bigl(\bigcup_iM_i\bigr)''\) generated by the union is injective.

Proof. (a) Let \(N=\bigcap_iM_i\) and let \(E_i\colon B(H)\to M_i\) be ucp projections. For each \(x\), \(E_i(x)\) lies in the ball of radius \(\|x\|\) of \(B(H)\), which is σ-weakly compact. By Tychonoff's theorem the net \((E_i)\) has a subnet \((E_{i(\lambda)})\) converging pointwise σ-weakly to a map \(E\colon B(H)\to B(H)\). Pointwise limits of ucp maps are linear, unital and completely positive, since the positive cone of each \(M_n(B(H))\) is σ-weakly closed. Fix \(j\). For \(i(\lambda)\ge j\) we have \(E_{i(\lambda)}(x)\in M_{i(\lambda)}\subset M_j\), and \(M_j\) is σ-weakly closed, so \(E(x)\in M_j\). Hence \(E(x)\in N\). If \(x\in N\) then \(E_i(x)=x\) for all \(i\), so \(E(x)=x\). Thus \(E\) is a ucp projection onto \(N\), and \(N\) is injective by Theorem 2.1.

(b) By the double commutant theorem \(M'=\bigl(\bigcup_iM_i\bigr)'=\bigcap_iM_i'\). The family \((M_i')\) is decreasing, and each \(M_i'\) is injective by Theorem 4.2. By (a), \(M'\) is injective, and by Theorem 4.2 again, so is \(M\). \(\square\)

Note that in (b) the direct argument, taking a limit of projections onto the \(M_i\), produces a ucp map \(E\) with values in \(M\) that is the identity on \(\bigcup_iM_i\), but not obviously on the weak closure: \(E\) need not be normal. The passage through commutants avoids this difficulty. Exercise 8.2 shows that projections of norm one are often not normal.

5. Averaging over amenable groups

A reference for this section is [Connes 1976, §4.2 and §4.4].

Let \(N\subset B(H)\) be a von Neumann algebra. A unitary \(u\in B(H)\) normalizes \(N\) if \(uNu^*=N\).

Theorem 5.1. Let \(N\subset B(H)\) be an injective von Neumann algebra, \(G\) an amenable locally compact group and \(u\colon G\to B(H)\) a strongly continuous unitary representation such that every \(u(g)\) normalizes \(N\). Then the von Neumann algebra \(M\) generated by \(N\) and \(u(G)\) is injective.

In particular, if \(\mathcal G\) is a group of unitaries normalizing \(N\) and \(\mathcal G\) is amenable as a discrete group, then the von Neumann algebra generated by \(N\) and \(\mathcal G\) is injective (take the inclusion \(u\colon\mathcal G\to B(H)\), which is continuous for the discrete topology).

Proof. By the double commutant theorem, \(M'=N'\cap u(G)'\). By Theorem 4.2 it suffices to show that \(M'\) is injective. By Theorem 4.2 again, \(N'\) is injective; let \(E_0\colon B(H)\to N'\) be a ucp projection. Let \(m\) be a left-invariant mean on \(C_b(G)\) (B8).

For \(y\in N'\) and \(\xi,\eta\in H\), the function \(f_{\xi,\eta}(g)=\langle u(g)yu(g)^*\xi,\eta\rangle\) is bounded by \(\|y\|\|\xi\|\|\eta\|\), and it is continuous: \(g\mapsto yu(g^{-1})\xi\) is norm continuous, and if \(z_g\to z_{g_0}\) in norm then \(u(g)z_g\to u(g_0)z_{g_0}\), because \(u\) is strongly continuous and unitary. The form \((\xi,\eta)\mapsto m(f_{\xi,\eta})\) is sesquilinear and bounded by \(\|y\|\|\xi\|\|\eta\|\), so there is a unique \(P(y)\in B(H)\) with \[ \langle P(y)\xi,\eta\rangle=m\bigl(g\mapsto\langle u(g)yu(g)^*\xi,\eta\rangle\bigr),\qquad\|P(y)\|\le\|y\|. \tag{5.1} \] We check the properties of \(P\colon N'\to B(H)\).

Values in \(N'\). Let \(a\in N\). Since \(u(g)^*au(g)\in N\) and \(y\in N'\), we get \(u(g)yu(g)^*a=u(g)y\bigl(u(g)^*au(g)\bigr)u(g)^*=a\,u(g)yu(g)^*\). Hence \(\langle P(y)a\xi,\eta\rangle=m(\langle u(g)yu(g)^*\xi,a^*\eta\rangle)=\langle P(y)\xi,a^*\eta\rangle=\langle aP(y)\xi,\eta\rangle\).

Values in \(u(G)'\). For \(h\in G\), \(\langle u(h)P(y)u(h)^*\xi,\eta\rangle=\langle P(y)u(h)^*\xi,u(h)^*\eta\rangle=m\bigl(g\mapsto\langle u(hg)yu(hg)^*\xi,\eta\rangle\bigr)=m(f_{\xi,\eta})\) by left invariance. So \(u(h)P(y)u(h)^*=P(y)\).

Identity on \(M'\). If \(y\in M'\) then \(u(g)yu(g)^*=y\) for all \(g\), so \(f_{\xi,\eta}\) is constant and \(P(y)=y\).

Complete positivity. For \(y=(y_{kl})\in M_n(N')\) and \(\xi=(\xi_l)\in H^n\), \[ \langle P_n(y)\xi,\xi\rangle=\sum_{k,l}m\bigl(\langle u(g)y_{kl}u(g)^*\xi_l,\xi_k\rangle\bigr) =m\bigl(g\mapsto\langle(u(g)\otimes1_n)y(u(g)\otimes1_n)^*\xi,\xi\rangle\bigr), \] which is \(\ge0\) when \(y\ge0\), because \(m\) is positive. Also \(P(1)=1\).

So \(P\circ E_0\colon B(H)\to M'\) is ucp and the identity on \(M'\). By Theorem 2.1, \(M'\) is injective. \(\square\)

Corollary 5.2.

(a) Every abelian von Neumann algebra is injective.

(b) Every von Neumann algebra of type I is injective.

(c) Every approximately finite-dimensional von Neumann algebra is injective. Here \(M\subset B(H)\) is approximately finite-dimensional if \(M=\bigl(\bigcup_iA_i\bigr)''\) for an increasing family, indexed by a directed set, of finite-dimensional \(\ast\)-subalgebras \(A_i\) containing \(1_H\).

Proof. (a) Let \(A\subset B(H)\) be abelian and \(\mathcal U(A)\) its unitary group. A C*-algebra is spanned by its unitaries, so \(A=\mathcal U(A)''\). The group \(\mathcal U(A)\) is abelian, hence amenable as a discrete group (B8), and it normalizes \(\mathbb C1_H\), which is injective (Example 2.4). By Theorem 5.1, \(A\) is injective.

(b) By (B7) a type I algebra is isomorphic to \(\prod_jA_j\bar\otimes B(K_j)\) with \(A_j\) abelian. Each \(A_j\) is injective by (a), each \(A_j\bar\otimes B(K_j)\) by Proposition 2.3(c), and the product by Proposition 2.3(b).

(c) Each \(A_i\) is a finite-dimensional C*-algebra, hence a von Neumann algebra in \(H\), and injective by Example 2.4. Apply Theorem 4.3(b). \(\square\)

Corollary 5.3. Let \(\pi\) be a strongly continuous unitary representation of an amenable locally compact group \(G\). Then \(\pi(G)''\) and \(\pi(G)'\) are injective.

Proof. Apply Theorem 5.1 with \(N=\mathbb C1\), which is injective and normalized by every unitary. This gives \(\pi(G)''\); then Theorem 4.2 gives \(\pi(G)'\). \(\square\)

The continuity of \(\pi\) matters in the proof. The compact group \(SO(3)\) is amenable, but it is not amenable as a discrete group (B8), so Theorem 5.1 with the discrete topology does not apply to it; Corollary 5.3 does, through the continuity of the orbit maps \(g\mapsto\langle\pi(g)y\pi(g)^*\xi,\eta\rangle\).

Crossed products by discrete groups. Let \(N\subset B(H)\) be a von Neumann algebra, \(\Gamma\) a discrete group and \(\alpha\colon\Gamma\to\operatorname{Aut}N\) an action. On \(\ell^2(\Gamma,H)\) define \[ (\pi_\alpha(a)\xi)(h)=\alpha_{h^{-1}}(a)\xi(h),\qquad(\lambda(g)\xi)(h)=\xi(g^{-1}h)\qquad(a\in N,\ g,h\in\Gamma). \] Then \(\pi_\alpha\) is the direct sum of the faithful normal representations \(\alpha_{h^{-1}}\), so it is faithful and normal and \(\pi_\alpha(N)\) is a von Neumann algebra isomorphic to \(N\) (B5). A direct computation gives \[ \bigl(\lambda(g)\pi_\alpha(a)\lambda(g)^*\xi\bigr)(h)=\alpha_{h^{-1}g}(a)\xi(h)=\alpha_{h^{-1}}(\alpha_g(a))\xi(h), \] so \(\lambda(g)\pi_\alpha(a)\lambda(g)^*=\pi_\alpha(\alpha_g(a))\) and each \(\lambda(g)\) normalizes \(\pi_\alpha(N)\). The crossed product \(N\rtimes_\alpha\Gamma\) is the von Neumann algebra generated by \(\pi_\alpha(N)\) and \(\lambda(\Gamma)\).

When \((X,\mu)\) is a σ-finite measure space and \(\Gamma\) acts on it by measurable bijections \(T_g\) that preserve the null sets, \(\alpha_g(f)=f\circ T_g^{-1}\) defines an action on \(L^\infty(X,\mu)\) (acting on \(L^2(X,\mu)\) by multiplication). The crossed product \(L^\infty(X,\mu)\rtimes_\alpha\Gamma\) is the group measure space construction.

Corollary 5.4.

(a) If \(N\) is injective and \(\Gamma\) is an amenable discrete group, then \(N\rtimes_\alpha\Gamma\) is injective.

(b) For an action of an amenable discrete group, the group measure space construction is injective.

Proof. (a) The algebra \(\pi_\alpha(N)\cong N\) is injective, and \(\lambda\colon\Gamma\to B(\ell^2(\Gamma,H))\) is a unitary representation of the amenable group \(\Gamma\) normalizing it. Apply Theorem 5.1. (b) is (a) with \(N=L^\infty(X,\mu)\), which is injective by Corollary 5.2(a). \(\square\)

The same proof applies to the crossed product of a von Neumann algebra by a continuous action of an amenable locally compact group \(G\): that crossed product is generated by a faithful normal image of \(N\) and a strongly continuous unitary representation of \(G\) normalizing it (Changing the Hilbert space of a regular crossed product, OA-FLOW.REG.CONSTRUCTION–REG.NORMALITY), so Theorem 5.1 applies.

For a discrete group \(\Gamma\), the group von Neumann algebra is \(L(\Gamma)=\lambda(\Gamma)''\subset B(\ell^2\Gamma)\), where \(\lambda\) is the left regular representation. Write \(\delta_g\) for the standard basis of \(\ell^2\Gamma\).

Theorem 5.5. For a discrete group \(\Gamma\), the following are equivalent: (i) \(\Gamma\) is amenable; (ii) \(L(\Gamma)\) is injective; (iii) \(L(\Gamma)'\) is injective.

Proof. (i)⇒(ii) is Corollary 5.3, and (ii)⇔(iii) is Theorem 4.2.

(ii)⇒(i). Let \(\tau(x)=\langle x\delta_e,\delta_e\rangle\) on \(L(\Gamma)\). It is a state, and it is a trace: \(\tau(\lambda(g)\lambda(h))=\tau(\lambda(h)\lambda(g))\) since both are \(1\) if \(gh=e\) and \(0\) otherwise, so \(\tau(xy)=\tau(yx)\) for \(x,y\) in the group algebra \(\mathbb C\Gamma\). The group algebra is a unital \(\ast\)-algebra, so by the double commutant theorem it is weak operator dense in \(L(\Gamma)\). For fixed \(y\), both \(x\mapsto\tau(xy)\) and \(x\mapsto\tau(yx)\) are weak operator continuous; so the identity extends first to \(x\in L(\Gamma)\), \(y\in\mathbb C\Gamma\), and then, in the same way, to all \(x,y\in L(\Gamma)\).

Let \(E\colon B(\ell^2\Gamma)\to L(\Gamma)\) be a projection of norm one; by Theorem 2.1 it is ucp and \(L(\Gamma)\)-bimodular. For \(f\in\ell^\infty(\Gamma)\) let \(M_f\) be the multiplication operator \(M_f\delta_h=f(h)\delta_h\). Then \(\lambda(g)M_f\lambda(g)^*\delta_h=f(g^{-1}h)\delta_h\), that is, \(\lambda(g)M_f\lambda(g)^*=M_{f(g^{-1}\,\cdot\,)}\). Define \(m(f)=\tau(E(M_f))\). This is a positive linear functional with \(m(1)=1\), hence a mean on \(\ell^\infty(\Gamma)\), and \[ m\bigl(f(g^{-1}\,\cdot\,)\bigr)=\tau\bigl(E(\lambda(g)M_f\lambda(g)^*)\bigr)=\tau\bigl(\lambda(g)E(M_f)\lambda(g)^*\bigr) =\tau(E(M_f))=m(f). \] As \(g\) runs over \(\Gamma\) so does \(g^{-1}\), so \(m\) is left-invariant and \(\Gamma\) is amenable. \(\square\)

Example 5.6. The free group \(\mathbb F_2\) is not amenable (B8), so \(L(\mathbb F_2)\) and its commutant are not injective. Thus \(B(\ell^2\mathbb F_2)\), which is injective, contains a von Neumann subalgebra that is not: injectivity does not pass to subalgebras. For a subalgebra \(N\subset M\) with \(M\) injective, \(N\) is injective exactly when there is a projection of norm one from \(M\) onto \(N\) (compose with a projection \(B(H)\to M\), and use Theorem 2.1).

For locally compact groups that are not discrete, the implication (ii)⇒(i) fails: the group \(SL(2,\mathbb R)\) is not amenable, yet every representation of it generates an injective von Neumann algebra (Proposition 7.2(b)).

6. Direct integrals

A reference for this section is [Connes 1976, §4].

Throughout this section we fix a separable Hilbert space \(H\), a probability measure \(\mu\) on a standard Borel space \(X\), and a Borel assignment \(\alpha\mapsto M(\alpha)\) of von Neumann algebras in \(H\) to the points of \(X\), with bounded Borel fields \(a_k\colon X\to B_1\) such that \(\{a_k(\alpha)\}_k\) is strong* dense in the unit ball of \(M(\alpha)\) for every \(\alpha\) (B10). Let \(D\) be the algebra of decomposable operators on \(L^2(X,\mu;H)\) and \(M=\int^\oplus_XM(\alpha)\,d\mu(\alpha)\).

Membership in \(M(\alpha)\) is a Borel condition. The unit ball of \(M(\alpha)\) is strong* closed, so for \(y\in B_1\), \[ y\in M(\alpha)\iff \inf_kd\bigl(y,a_k(\alpha)\bigr)=0, \tag{6.1} \] with \(d\) the metric of (B9), and \((\alpha,y)\mapsto\inf_kd(y,a_k(\alpha))\) is a Borel function on \(X\times B_1\).

The proof of the main theorem needs a form of the failure of injectivity that can be tested on countably many elements. For each \(n\) fix a countable dense subset \(\Sigma_n\) of \((M_n)_{sa}\).

Definition 6.1. Let \(N\subset B(K)\) be a von Neumann algebra, \(n,q,k\ge1\) integers and \(\sigma\in(M_n)_{sa}\). A witness for \((n,\sigma,q,k)\) in \(N\) is a triple \((b,s,\rho)\) with

• \(b\in B(K)_{sa}\), \(\|b\|\le1\);
• \(s\in M_n(N)_{sa}\), \(\|s\|\le k\), and \(b\otimes\sigma\le s\);
• \(\rho\) a positive trace-class operator on \(K\otimes\mathbb C^n\) with \(\operatorname{Tr}\rho\le k\), such that \[ \operatorname{Tr}\bigl(\rho(x\otimes\sigma)\bigr)\ge\operatorname{Tr}(\rho s)+\tfrac1q\qquad\text{for all }x\in N_{sa},\ \|x\|\le1. \tag{6.2} \]

Lemma 6.2. Let \(N\subset B(K)\) be a von Neumann algebra.

(a) If there is a witness in \(N\) for some \((n,\sigma,q,k)\), then \(N\) is not injective.

(b) If \(N\) is not injective, there is a witness in \(N\) for some \((n,\sigma,q,k)\) with \(\sigma\in\Sigma_n\).

Proof. (a) If \(N\) were injective, Theorem 3.4 would give \(x\in N_{sa}\) with \(\|x\|\le\|b\|\le1\) and \(x\otimes\sigma\le s\), hence \(\operatorname{Tr}(\rho(x\otimes\sigma))\le\operatorname{Tr}(\rho s)\), against (6.2).

(b) By Theorem 3.4 there are \(b_0\in B(K)_{sa}\) and a \(b_0\)-constraint \((n,s_0,\sigma_0)\) without solution. Then \(b_0\ne0\), since \(x=0\) solves every \(0\)-constraint. Put \(b=b_0/\|b_0\|\) and \(s_1=s_0/\|b_0\|\); then \(b\otimes\sigma_0\le s_1\) and no \(x\in N_{sa}\) with \(\|x\|\le1\) satisfies \(x\otimes\sigma_0\le s_1\).

Work in the real vector space \(Z=M_n(B(K))_{sa}\) with the σ-weak topology; its continuous linear functionals are \(z\mapsto\operatorname{Tr}(\rho z)\) with \(\rho\) self-adjoint of trace class (B4). The set \(C=\{s_1-x\otimes\sigma_0:x\in N_{sa},\ \|x\|\le1\}\) is convex and σ-weakly compact, as the image of the σ-weakly compact self-adjoint unit ball of \(N\) under a continuous affine map. The positive cone \(Z_+\) is convex and σ-weakly closed, and \(C\cap Z_+=\emptyset\). By the Hahn–Banach separation theorem there are a self-adjoint trace-class \(\rho\) and \(\varepsilon>0\) with \(\operatorname{Tr}(\rho c)+\varepsilon\le\operatorname{Tr}(\rho p)\) for \(c\in C\), \(p\in Z_+\) (the supremum over the compact set \(C\) is attained). Since \(Z_+\) is a cone containing \(0\), this forces \(\operatorname{Tr}(\rho p)\ge0\) for all \(p\ge0\), so \(\rho\ge0\), and also \(\operatorname{Tr}(\rho c)\le-\varepsilon\) for \(c\in C\). That is, \[ \operatorname{Tr}(\rho(x\otimes\sigma_0))\ge\operatorname{Tr}(\rho s_1)+\varepsilon\qquad(x\in N_{sa},\ \|x\|\le1). \] Now perturb \(\sigma_0\). Choose \(\delta>0\) with \(2\delta\operatorname{Tr}\rho\le\varepsilon/2\), then \(\sigma\in\Sigma_n\) with \(\|\sigma-\sigma_0\|\le\delta\), and put \(s=s_1+\delta1\). Then \(b\otimes\sigma\le b\otimes\sigma_0+\|b\|\delta\le s\). For \(x\in N_{sa}\) with \(\|x\|\le1\), since \(\|x\otimes(\sigma-\sigma_0)\|\le\delta\) and \(|\operatorname{Tr}(\rho z)|\le\|z\|\operatorname{Tr}\rho\), \[ \operatorname{Tr}(\rho(x\otimes\sigma))\ge\operatorname{Tr}(\rho s_1)+\varepsilon-\delta\operatorname{Tr}\rho =\operatorname{Tr}(\rho s)+\varepsilon-2\delta\operatorname{Tr}\rho\ge\operatorname{Tr}(\rho s)+\varepsilon/2 . \] Choose integers \(q\ge2/\varepsilon\) and \(k\ge\max(\|s\|,\operatorname{Tr}\rho)\). Then \((b,s,\rho)\) is a witness for \((n,\sigma,q,k)\). \(\square\)

Lemma 6.3. Fix \(n,q,k\) and \(\sigma\in(M_n)_{sa}\). Let \(Y\) be the product of the self-adjoint part of the unit ball of \(B(H)\), the self-adjoint part of the ball of radius \(k\) of \(B(H\otimes\mathbb C^n)\) (both with the strong* topology) and the positive trace-class operators on \(H\otimes\mathbb C^n\) of trace at most \(k\) (with the trace norm). Then \(Y\) is a standard Borel space, and the set \(W\) of \((\alpha,b,s,\rho)\in X\times Y\) such that \((b,s,\rho)\) is a witness for \((n,\sigma,q,k)\) in \(M(\alpha)\) is a Borel subset of \(X\times Y\).

Proof. \(Y\) is a product of Polish spaces by (B9). A point \((\alpha,b,s,\rho)\) lies in \(W\) exactly when the following three conditions hold.

1. Each entry \(s_{ij}\) of \(s\) (a strong* continuous function of \(s\)) satisfies \(\inf_ld(s_{ij}/k,a_l(\alpha))=0\). By (6.1) and \(\|s_{ij}\|\le k\), this says \(s\in M_n(M(\alpha))\).
2. \(s-b\otimes\sigma\ge0\). This is a closed condition on \((b,s)\).
3. \(\operatorname{Tr}(\rho(\operatorname{Re}a_l(\alpha)\otimes\sigma))\ge\operatorname{Tr}(\rho s)+1/q\) for every \(l\), where \(\operatorname{Re}a=(a+a^*)/2\).

Each condition defines a Borel set: the maps \(\alpha\mapsto\operatorname{Re}a_l(\alpha)\) are Borel, and the pairing \((\rho,z)\mapsto\operatorname{Tr}(\rho z)\) is continuous on the product of the trace-class operators of trace at most \(k\) (trace norm) and a ball of \(B(H\otimes\mathbb C^n)\) (weak operator topology), because \(|\operatorname{Tr}(\rho'z')-\operatorname{Tr}(\rho z)|\le\|\rho'-\rho\|_1\|z'\|+|\operatorname{Tr}(\rho(z'-z))|\).

It remains to see that condition 3 is equivalent to (6.2) for \(N=M(\alpha)\). The map \(a\mapsto\operatorname{Re}a\) is strong* continuous and maps the unit ball of \(M(\alpha)\) onto its self-adjoint unit ball (it fixes self-adjoint elements), so \(\{\operatorname{Re}a_l(\alpha)\}_l\) is strong* dense in the self-adjoint unit ball of \(M(\alpha)\). The function \(x\mapsto\operatorname{Tr}(\rho(x\otimes\sigma))\) is continuous for the weak operator topology on bounded sets, hence for the strong* topology. So an inequality valid on the dense set holds on the whole self-adjoint unit ball. \(\square\)

Theorem 6.4. The set of \(\alpha\in X\) for which \(M(\alpha)\) is injective is \(\mu\)-measurable, and \(M=\int^\oplus_XM(\alpha)\,d\mu(\alpha)\) is injective if and only if \(M(\alpha)\) is injective for \(\mu\)-almost every \(\alpha\).

Reference: [Connes 1976, §4].

Proof. For each \((n,\sigma,q,k)\) with \(\sigma\in\Sigma_n\), let \(W_{n,\sigma,q,k}\) be the Borel set of Lemma 6.3 and \(A_{n,\sigma,q,k}\) its projection to \(X\). By Lemma 6.2, \(M(\alpha)\) fails to be injective exactly when \(\alpha\) lies in the countable union \(B\) of the sets \(A_{n,\sigma,q,k}\). By (B11), \(B\) is analytic, hence \(\mu\)-measurable.

\(M\) injective implies \(\mu(B)=0\). Suppose \(M\) is injective and \(\mu(B)>0\). Then some \(A=A_{n,\sigma,q,k}\) has \(\mu(A)>0\). By (B11) there are a Borel set \(A_0\subset A\) with \(\mu(A_0)=\mu(A)>0\) and a Borel map \(\alpha\mapsto(b_\alpha,s_\alpha,\rho_\alpha)\) on \(A_0\) such that \((b_\alpha,s_\alpha,\rho_\alpha)\) is a witness for \((n,\sigma,q,k)\) in \(M(\alpha)\) for every \(\alpha\in A_0\). Put \(b_\alpha=0\) and \(s_\alpha=0\) for \(\alpha\notin A_0\). The fields \(\alpha\mapsto b_\alpha\) and \(\alpha\mapsto s_\alpha\) are bounded and Borel, so they define \(b\in D_{sa}\) with \(\|b\|\le1\) and \(s\in M_n(D)_{sa}\). The entries of \(s_\alpha\) lie in \(M(\alpha)\) for every \(\alpha\), so \(s\in M_n(M)\). The field of \(s-b\otimes\sigma\) is positive everywhere, so \(b\otimes\sigma\le s\) (B10). Since \(M\) is injective, Theorem 3.4, applied in \(B(L^2(X,\mu;H))\), gives \(x\in M_{sa}\) with \(\|x\|\le\|b\|\le1\) and \(x\otimes\sigma\le s\). Choose a Borel field for \(x\). For almost every \(\alpha\) we then have \(x(\alpha)\in M(\alpha)\), \(x(\alpha)=x(\alpha)^*\), \(\|x(\alpha)\|\le1\) and \(x(\alpha)\otimes\sigma\le s_\alpha\) (B10). Since \(\mu(A_0)>0\), there is such an \(\alpha\) in \(A_0\). For it, positivity of \(\rho_\alpha\) gives \(\operatorname{Tr}(\rho_\alpha(x(\alpha)\otimes\sigma))\le\operatorname{Tr}(\rho_\alpha s_\alpha)\), while (6.2) gives \(\operatorname{Tr}(\rho_\alpha(x(\alpha)\otimes\sigma))\ge\operatorname{Tr}(\rho_\alpha s_\alpha)+1/q\). This is a contradiction.

\(\mu(B)=0\) implies \(M\) injective. We verify condition (ii) of Theorem 3.4 for \(M\subset B(L^2(X,\mu;H))\). First, \(D=(L^\infty(X,\mu)\otimes1_H)'\) is the commutant of an abelian von Neumann algebra, hence injective by Corollary 5.2(a) and Theorem 4.2; let \(F\colon B(L^2(X,\mu;H))\to D\) be a ucp projection. Let \(b\in B(L^2(X,\mu;H))_{sa}\) and let \((n,s,\sigma)\) be a \(b\)-constraint for \(M\). Put \(r=\|b\|\). If \(r=0\), then \(x=0\) is a solution. Otherwise let \(b'=F(b)\). Then \(b'\in D_{sa}\), \(\|b'\|\le r\), and, since \(F_n\) is positive and the identity on \(M_n(M)\subset M_n(D)\), \[ s-b'\otimes\sigma=F_n(s-b\otimes\sigma)\ge0 . \] Choose Borel fields for \(b'\) and for \(s\). There is a Borel set \(X_1\) with \(\mu(X\setminus X_1)=0\) such that for \(\alpha\in X_1\): \(M(\alpha)\) is injective, \(b'(\alpha)\) is self-adjoint with \(\|b'(\alpha)\|\le r\), \(s(\alpha)\) is a self-adjoint element of \(M_n(M(\alpha))\), and \(b'(\alpha)\otimes\sigma\le s(\alpha)\). Let \(G\) be the set of pairs \((\alpha,y)\in X_1\times B_r\) with \(y=y^*\), \(y/r\in M(\alpha)\) and \(s(\alpha)-y\otimes\sigma\ge0\). By (6.1) and (B9), \(G\) is Borel. For \(\alpha\in X_1\), the constraint \((n,s(\alpha),\sigma)\) for the injective algebra \(M(\alpha)\) with the operator \(b'(\alpha)\) has a solution by Theorem 3.4, and every solution \(y\) gives \((\alpha,y)\in G\), since \(\|y\|\le\|b'(\alpha)\|\le r\). So the projection of \(G\) is \(X_1\). By (B11) there is a Borel set \(X_2\subset X_1\) with \(\mu(X\setminus X_2)=0\) and a Borel map \(y\colon X_2\to B_r\) with \((\alpha,y(\alpha))\in G\). Put \(y(\alpha)=0\) off \(X_2\) and let \(x\in D\) be the operator of the field \(y\). Then \(x=x^*\), \(\|x\|\le r=\|b\|\), \(x\in M\) because \(y(\alpha)\in M(\alpha)\) almost everywhere, and \(x\otimes\sigma\le s\) because the field of \(s-x\otimes\sigma\) is positive almost everywhere (B10). So \(x\) solves the constraint, and \(M\) is injective by Theorem 3.4. \(\square\)

Example 6.5. Take \(X=\{1,2\}\) with \(\mu(\{1\})=\mu(\{2\})=1/2\). Then \(L^2(X,\mu;H)=H\oplus H\) and \(M=M(1)\oplus M(2)\), and Theorem 6.4 says that \(M(1)\oplus M(2)\) is injective exactly when both summands are, which is Proposition 2.3(b) for two summands. With \(H=\ell^2\mathbb F_2\), \(M(1)=B(H)\) and \(M(2)=L(\mathbb F_2)\), the algebra \(M\) is not injective although half of it is.

Now take \(X=[0,1]\) with Lebesgue measure, \(M(\alpha)=B(H)\) for \(\alpha\ne\frac12\) and \(M(\frac12)=L(\mathbb F_2)\). Changing a field on a null set does not change the operator it defines, so \(M\) is the same algebra as for the constant field \(B(H)\), namely \(D\), which is injective. This is why the theorem can only speak of almost every fibre.

7. Representations of connected groups

A reference for this section is [Connes 1976, §4.3].

A locally compact group is of type I if \(\pi(G)''\) is a von Neumann algebra of type I for every strongly continuous unitary representation \(\pi\) of \(G\) on a separable Hilbert space.

Theorem 7.1. Let \(G\) be a connected locally compact group and \(\pi\) a strongly continuous unitary representation of \(G\) on a separable Hilbert space. Then \(\pi(G)''\) and \(\pi(G)'\) are injective.

Reference: [Connes 1976, §4.3], where the result is stated after Corollary 4.3.2 in the more general form of Corollary 7.5 below.

By Theorem 4.2 the two conclusions are equivalent. Some cases follow at once from Section 5.

Proposition 7.2. The conclusion of Theorem 7.1 holds, for any locally compact group \(G\) (connected or not), in each of the following cases.

(a) \(G\) is amenable. For connected groups this includes all solvable, nilpotent and compact connected groups, and extensions of such groups by one another.

(b) \(G\) is of type I. This includes all connected semisimple Lie groups and all connected nilpotent Lie groups (theorems of Harish-Chandra and of Dixmier; see [Dixmier 1957, Introduction]), and more generally every connected Lie group whose radical is nilpotent ((L3) below).

(c) \(\pi\) is a direct sum of representations \(\pi_j\) with \(\pi_j(G)''\) injective and pairwise disjoint (no nonzero intertwiner between \(\pi_j\) and \(\pi_k\) for \(j\ne k\)).

Proof. (a) is Corollary 5.3; solvable groups are amenable, compact groups are amenable, and extensions of amenable groups by amenable groups are amenable ([Bekka–de la Harpe–Valette, Proposition G.2.2 and Corollary G.2.3]). (b) follows from Corollary 5.2(b). (c) An operator commuting with \(\pi(G)\) is a matrix \((T_{jk})\) of operators with \(T_{jk}\pi_k(g)=\pi_j(g)T_{jk}\); by disjointness \(T_{jk}=0\) for \(j\ne k\). So \(\pi(G)'=\prod_j\pi_j(G)'\) (block diagonal), and taking commutants, \(\pi(G)''=\prod_j\pi_j(G)''\). Apply Proposition 2.3(b). \(\square\)

Three results on Lie groups. The general case uses three results from the structure theory of locally compact groups and Lie groups. (L1) is proved below. (L2) and (L3) are quoted; their proofs belong to the structure theory of Lie groups and are not given in this course.

(L1) Lie quotients of connected groups. Let \(L\) be a connected locally compact group and \(U\) a neighbourhood of the identity. There is a compact normal subgroup \(K\) of \(L\) with \(K\subset U\) such that \(L/K\) is isomorphic to a connected Lie group.

This is the case of connected groups of the Gleason–Yamabe theorem [Tao 2012, Theorem 1.1.17], and it is the only case that Section 7 uses. It is proved after (L3).

(L2) A closed normal subgroup with abelian quotient. Let \(L\) be a connected Lie group with Lie algebra \(\mathfrak l\), and let \(\mathfrak n\) be the largest nilpotent ideal of \(\mathfrak l\). The connected Lie subgroup \(D\) of \(L\) with Lie algebra \([\mathfrak l,\mathfrak l]+\mathfrak n\) is closed and normal in \(L\), the quotient \(L/D\) is abelian, and the radical of \(D\) is nilpotent [Dixmier 1969, Proposition 1.7]. That \(D\) is closed is a theorem of Pukánszky, proved in [Dixmier 1969, Proposition 1.5].

(L3) A type I criterion. A connected Lie group whose radical is nilpotent is of type I [Dixmier 1969, Proposition 2.3(i)]. Dixmier calls a group of type I when every strongly continuous unitary representation, on any Hilbert space, generates a von Neumann algebra of type I [Dixmier 1957, Introduction]; this implies the definition above.

No condition on the centre enters (L3); it applies, for instance, to every connected semisimple Lie group, whose radical is zero. Being of type I is not a property of the Lie algebra alone: there are two locally isomorphic connected Lie groups of which one is of type I and the other is not [Dixmier 1969, Remark 2.2].

Proof of (L1). Three facts on a locally compact group \(L\) and a closed subgroup \(H\) are proved in Quotient measures and Weil's integration formula, Lemma 1.2: the coset space \(L/H\) is locally compact and Hausdorff, the quotient map \(q:L\to L/H\) is open, and every compact subset of \(L/H\) is the image of a compact subset of \(L\). If \(H\) is normal, \(L/H\) is a topological group: the map \(q\times q\) is open and surjective, hence a quotient map, and the group operations of \(L/H\) composed with \(q\times q\) and with \(q\) are continuous.

(L1a) A connected locally compact group \(L\) is \(\sigma\)-compact. Let \(V\) be a compact neighbourhood of the identity with \(V=V^{-1}\); for instance \(V'\cap V'^{-1}\) for a compact neighbourhood \(V'\). Each \(V^n\) is compact, as the image of \(V\times\dots\times V\) under multiplication. The union \(L'=\bigcup_nV^n\) is a subgroup. It is open, because for \(x\in L'\) the neighbourhood \(xV\) of \(x\) lies in \(L'\). An open subgroup is closed, its complement being a union of open cosets. Since \(L\) is connected, \(L'=L\).

(L1b) (Kakutani–Kodaira.) Let \(L\) be \(\sigma\)-compact and \(U\) a neighbourhood of the identity. There is a compact normal subgroup \(K_1\subset U\) such that \(L/K_1\) is second countable. Write \(L=\bigcup_mC_m\) with compact sets \(C_1\subset C_2\subset\cdots\). For a compact set \(C\) and a neighbourhood \(V\) of the identity, there is a neighbourhood \(W\) of the identity with \(cWc^{-1}\subset V\) for every \(c\in C\). Indeed, the map \((c,x)\mapsto cxc^{-1}\) is continuous and sends \(C\times\{1\}\) into the interior of \(V\), so each \(c\in C\) has neighbourhoods \(O_c\) of \(c\) and \(W_c\) of the identity with \(O_c\times W_c\) mapped into \(V\). Finitely many \(O_c\) cover \(C\), and \(W\) is the intersection of the corresponding \(W_c\). So we can choose compact neighbourhoods \(V_1\supset V_2\supset\cdots\) of the identity with \(V_n=V_n^{-1}\), \(V_1\subset U\), \(V_{n+1}V_{n+1}\subset V_n\) and \(cV_{n+1}c^{-1}\subset V_n\) for \(c\in C_n\).

Put \(K_1=\bigcap_nV_n\). It is compact and contained in \(U\). It is a subgroup, because \(V_{n+1}V_{n+1}\subset V_n\) and every \(V_n\) is symmetric. It is normal: if \(g\in C_m\), then \(gV_{n+1}g^{-1}\subset V_n\) for every \(n\ge m\), so \(gK_1g^{-1}\subset\bigcap_{n\ge m}V_n=K_1\). Every open set \(O\) containing \(K_1\) contains some \(V_n\), because the compact sets \(V_n\setminus O\) decrease and have empty intersection.

Let \(q:L\to L/K_1\) be the quotient map and \(V_n^\circ\) the interior of \(V_n\). The sets \(q(V_n^\circ)\) are open neighbourhoods of the identity. Every neighbourhood \(N\) of the identity of \(L/K_1\) contains some \(q(V_n)\), since \(q^{-1}(N)\) contains an open set containing \(K_1\). For each pair \(m,n\), cover the compact set \(q(C_m)\) by finitely many translates \(x\,q(V_n^\circ)\) with \(x\in q(C_m)\), and let \(D_{m,n}\) be the finite set of the \(x\) used. The countable family of the sets \(x\,q(V_n^\circ)\) with \(x\in D_{m,n}\) is a base of the topology of \(L/K_1\). Let \(O\) be open and \(y\in O\). Choose \(n\) with \(q(V_n)\subset y^{-1}O\) and \(m\) with \(y\in q(C_m)\). Then \(y\in x\,q(V_{n+1}^\circ)\) for some \(x\in D_{m,n+1}\). So \(x\in y\,q(V_{n+1})\), because \(V_{n+1}\) is symmetric. Hence \[ x\,q(V_{n+1}^\circ)\subset y\,q(V_{n+1}V_{n+1})\subset y\,q(V_n)\subset O . \]

(L1c) Let \(K_1\) be a compact normal subgroup of \(L\), \(q:L\to L/K_1\) the quotient map, and \(F\) a compact subset of \(L/K_1\). Then \(q^{-1}(F)\) is compact. By the lemma quoted above, \(F=q(E)\) for a compact set \(E\subset L\). Then \(q^{-1}(F)=EK_1\), the image of the compact set \(E\times K_1\) under multiplication.

Proof of (L1). Choose a neighbourhood \(U'\) of the identity with \(U'U'\subset U\). By (L1a) and (L1b), there is a compact normal subgroup \(K_1\subset U'\) of \(L\) such that \(L/K_1\) is second countable. Let \(q:L\to L/K_1\) be the quotient map. The group \(L/K_1\) is locally compact, it is connected as an image of \(L\), and \(q(U')\) is a neighbourhood of its identity, because \(q\) is open. By Descent and the K-theory of crossed products, Theorem 12.31, applied to \(L/K_1\), there is a compact normal subgroup \(F\subset q(U')\) of \(L/K_1\) such that \((L/K_1)/F\) is a finite-dimensional Lie group. Put \(K=q^{-1}(F)\). It is a closed normal subgroup of \(L\), and it is compact by (L1c). It lies in \(U\): \(K\subset q^{-1}(q(U'))=U'K_1\subset U'U'\subset U\). The composite of the quotient maps \(L\to L/K_1\to(L/K_1)/F\) is a continuous, open and surjective homomorphism with kernel \(K\). So it induces a bijective homomorphism \(L/K\to(L/K_1)/F\), which is continuous and open. Thus \(L/K\) is isomorphic to a Lie group, and \(L/K\) is connected as an image of \(L\). \(\square\)

The theorem in (L1b) is due to Kakutani and Kodaira [Kakutani–Kodaira 1944].

The first lemma replaces the amenable group of Theorem 5.1 by an amenable quotient.

Lemma 7.3 (averaging over an amenable quotient). Let \(G\) be a locally compact group, \(K\) a closed normal subgroup of \(G\) such that \(G/K\) is amenable, and \(\pi\) a strongly continuous unitary representation of \(G\) on \(H\). If \(\pi(K)''\) is injective, then \(\pi(G)''\) and \(\pi(G)'\) are injective.

Proof. Put \(N=\pi(K)''\). For \(g\in G\), \(\pi(g)\pi(K)\pi(g)^*=\pi(gKg^{-1})=\pi(K)\), so \(\pi(g)\) normalizes \(N\). Theorem 5.1 does not apply as it stands, because \(G\) need not be amenable, and representatives of the cosets of \(K\) need not form a group of unitaries. We follow its proof, averaging over \(G/K\) instead of \(G\).

By Theorem 4.2, \(N'\) is injective; let \(E_0\colon B(H)\to N'\) be a ucp projection (Theorem 2.1). For \(y\in N'\) and \(\xi,\eta\in H\), the function \(f_{\xi,\eta}(g)=\langle\pi(g)y\pi(g)^*\xi,\eta\rangle\) on \(G\) is bounded by \(\|y\|\|\xi\|\|\eta\|\) and continuous, as shown in the proof of Theorem 5.1. It is constant on the cosets of \(K\): for \(k\in K\), \(\pi(k)\in N\) commutes with \(y\), so \(\pi(gk)y\pi(gk)^*=\pi(g)y\pi(g)^*\). Hence \(f_{\xi,\eta}=\tilde f_{\xi,\eta}\circ q\), where \(q\colon G\to G/K\) is the quotient map and \(\tilde f_{\xi,\eta}\) is a bounded function on \(G/K\). It is continuous: \(q\) is open, and \(\tilde f_{\xi,\eta}^{-1}(V)= q\bigl(f_{\xi,\eta}^{-1}(V)\bigr)\) for every open \(V\subset\mathbb C\). Let \(m\) be a left-invariant mean on \(C_b(G/K)\) (B8). As for (5.1), there is a unique \(P(y)\in B(H)\) with \[ \langle P(y)\xi,\eta\rangle=m\bigl(\tilde f_{\xi,\eta}\bigr),\qquad\|P(y)\|\le\|y\|. \] For \(h\in G\), \[ \langle\pi(h)P(y)\pi(h)^*\xi,\eta\rangle=m\bigl(\tilde f_{\pi(h)^*\xi,\pi(h)^*\eta}\bigr),\qquad \tilde f_{\pi(h)^*\xi,\pi(h)^*\eta}(gK)=\langle\pi(hg)y\pi(hg)^*\xi,\eta\rangle=\tilde f_{\xi,\eta}(hgK), \] so \(\tilde f_{\pi(h)^*\xi,\pi(h)^*\eta}\) is the left translate of \(\tilde f_{\xi,\eta}\) by \(hK\), and left invariance of \(m\) gives \(\pi(h)P(y)\pi(h)^*=P(y)\). That \(P\) takes values in \(N'\), is the identity on \(N'\cap\pi(G)'\), is completely positive and satisfies \(P(1)=1\) is checked word for word as in the proof of Theorem 5.1. Since \(N\subset\pi(G)''\), we have \(N'\cap\pi(G)'=\pi(G)'\). So \(P\circ E_0\) is a ucp projection of \(B(H)\) onto \(\pi(G)'\). By Theorem 2.1, \(\pi(G)'\) is injective, and by Theorem 4.2 so is \(\pi(G)''\). \(\square\)

The second lemma concerns the vectors fixed by a normal subgroup.

Lemma 7.4 (fixed vectors of a normal subgroup). Let \(\pi\) be a unitary representation of a group \(G\) on \(H\), \(M=\pi(G)''\), \(K\) a normal subgroup of \(G\), and \(p_K\) the projection onto \(H^K=\{\xi\in H:\pi(k)\xi=\xi\text{ for all }k\in K\}\).

(a) \(p_K\) is a central projection of \(M\).

(b) \(\|\xi-p_K\xi\|\le\sup_{k\in K}\|\pi(k)\xi-\xi\|\) for every \(\xi\in H\).

Proof. (a) Every \(x\in M'\) commutes with the operators \(\pi(k)\), so it maps \(H^K\) into itself, and so does \(x^*\in M'\). Hence \(p_Kx=xp_K\), and \(p_K\in M''=M\). If \(\xi\in H^K\), \(g\in G\) and \(k\in K\), then \(\pi(k)\pi(g)\xi=\pi(g)\pi(g^{-1}kg)\xi=\pi(g)\xi\), since \(g^{-1}kg\in K\). So \(\pi(g)H^K\subset H^K\) for every \(g\), with equality by applying this to \(g^{-1}\). Hence \(p_K\) commutes with \(\pi(G)\), that is, \(p_K\in M'\).

(b) Let \(C\) be the closed convex hull of the orbit \(\{\pi(k)\xi:k\in K\}\), and \(\eta\) the unique element of \(C\) of smallest norm. Each \(\pi(k)\) maps \(C\) onto itself and preserves norms, so \(\pi(k)\eta=\eta\) by uniqueness, and \(\eta\in H^K\). By (a), \(p_K\pi(k)\xi=\pi(k)p_K\xi=p_K\xi\), so the continuous linear map \(p_K\) is constant on the orbit, hence on \(C\), with value \(p_K\xi\). Thus \(\eta=p_K\eta=p_K\xi\). The closed ball with centre \(\xi\) and radius \(\sup_k\|\pi(k)\xi-\xi\|\) is convex and contains the orbit, so it contains \(C\), and in particular \(p_K\xi\). \(\square\)

Proof of Theorem 7.1. Step 1: connected Lie groups. Let \(G\) be a connected Lie group, and \(D\) the closed normal subgroup of (L2). By (L3), \(D\) is of type I, so \(\pi(D)''\) is of type I and therefore injective (Corollary 5.2(b)). The quotient \(G/D\) is abelian, hence amenable (B8). Lemma 7.3 with \(K=D\) shows that \(\pi(G)''\) and \(\pi(G)'\) are injective.

Step 2: central projections. Now let \(G\) be any connected locally compact group, and put \(M=\pi(G)''\). We may assume \(H\ne0\). For each neighbourhood \(U\) of the identity, (L1) gives a compact normal subgroup \(K_U\subset U\) of \(G\) such that \(G/K_U\) is a Lie group; it is connected, as a quotient of \(G\). Let \(p_U\) be the projection onto the vectors fixed by \(K_U\). By Lemma 7.4(a), \(p_U\) is a central projection of \(M\). Order the neighbourhoods of the identity by reverse inclusion. Then \(p_U\to1\) strongly: given \(\xi\in H\) and \(\varepsilon>0\), strong continuity gives a neighbourhood \(V\) of the identity with \(\|\pi(g)\xi-\xi\|\le\varepsilon\) for \(g\in V\), and for \(U\subset V\) Lemma 7.4(b) gives \(\|\xi-p_U\xi\|\le\varepsilon\). Applied to a unit vector with \(\varepsilon=1/2\), this shows that \(p_U\ne0\) for all \(U\) inside some neighbourhood \(V_0\); from now on \(U\) ranges over the neighbourhoods contained in \(V_0\), which are still directed by reverse inclusion.

Step 3: the corners are injective. On \(p_UH\) the representation \(\pi\) factors through \(G/K_U\): \(\pi_U(gK_U)=\pi(g)|_{p_UH}\) is well defined, since \(K_U\) acts trivially on \(p_UH\), and strongly continuous, since for \(\xi\in p_UH\) the map \(gK_U\mapsto\pi(g)\xi\) becomes continuous when composed with the open quotient map \(G\to G/K_U\). An operator \(T\) on \(p_UH\) that commutes with \(\pi_U(G/K_U)\), extended by \(0\) on \((1-p_U)H\), commutes with \(\pi(G)\), because \(p_U\) does; so it lies in \(M'\). Conversely \(M'\) maps \(p_UH\) into itself, since \(p_U\in M\), and its restrictions commute with \(\pi_U(G/K_U)\). Hence \(\pi_U(G/K_U)'=(M')_{p_U}\), and by (B5), applied to \(M'\) and the projection \(p_U\in(M')'\), \[ \pi_U(G/K_U)''=\bigl((M')_{p_U}\bigr)'=p_UMp_U|_{p_UH}=M_{p_U}. \] By Step 1 for the connected Lie group \(G/K_U\), this algebra is injective. By Theorem 2.1 there is a ucp projection \(E_U\) of \(B(p_UH)\) onto \(M_{p_U}\).

Step 4: a projection onto \(M\). Identify \(B(p_UH)\) with \(p_UB(H)p_U\); then \(M_{p_U}\) becomes \(p_UM\), which lies in \(M\). Fix a state \(\omega\) on \(B(H)\) and put \[ \Phi_U(x)=E_U(p_Uxp_U)+\omega(x)(1-p_U)\qquad(x\in B(H)). \] Both terms lie in \(M\), and both are completely positive in \(x\): the first is a compression followed by \(E_U\), and the second is a state times the positive element \(1-p_U\). Since \(\Phi_U(1)=p_U+(1-p_U)=1\), \(\Phi_U\colon B(H)\to M\) is ucp. For \(a\in M\), the element \(p_Uap_U=p_Ua\) lies in \(p_UM\) and is fixed by \(E_U\); since \(p_U\) is central, \[ \Phi_U(a)-a=p_Ua+\omega(a)(1-p_U)-a=(\omega(a)1-a)(1-p_U), \] so \(\|(\Phi_U(a)-a)\xi\|\le2\|a\|\,\|(1-p_U)\xi\|\to0\) for every \(\xi\in H\), by Step 2.

For each \(x\in B(H)\), the elements \(\Phi_U(x)\) lie in the ball of radius \(\|x\|\) of \(M\), which is σ-weakly compact (B4). By Tychonoff's theorem a subnet \((\Phi_{U_j})\) converges pointwise σ-weakly to a map \(E\colon B(H)\to M\). It is linear and unital. It is completely positive: if \((x_{kl})\in M_n(B(H))\) is positive and \(\xi_1,\dots,\xi_n\in H\), then \(\sum_{k,l}\langle E(x_{kl})\xi_l,\xi_k\rangle=\lim_j\sum_{k,l}\langle \Phi_{U_j}(x_{kl})\xi_l,\xi_k\rangle\ge0\). For \(a\in M\), \(\Phi_U(a)\to a\) strongly and boundedly, hence σ-weakly (B4), so \(E(a)=a\). Thus \(E\) is a ucp projection of \(B(H)\) onto \(M\), and \(M\) is injective by Theorem 2.1. By Theorem 4.2, \(\pi(G)'\) is injective as well. \(\square\)

Corollary 7.5. Let \(G\) be a locally compact group whose quotient \(G/G_0\) by the identity component \(G_0\) is amenable, and \(\pi\) a strongly continuous unitary representation of \(G\) on a separable Hilbert space. Then \(\pi(G)''\) and \(\pi(G)'\) are injective.

Proof. The identity component \(G_0\) is a closed subgroup, and it is normal, since each conjugation \(x\mapsto gxg^{-1}\) is a homeomorphism fixing the identity. It is a connected locally compact group, so \(\pi(G_0)''\) is injective by Theorem 7.1. Apply Lemma 7.3 with \(K=G_0\). \(\square\)

This is the form of the result stated in [Connes 1976, §4.3]. Case (b) of Proposition 7.2 already shows that the converse of Corollary 5.3 fails for non-discrete groups: \(SL(2,\mathbb R)\) is not amenable, yet all its representations generate injective von Neumann algebras, since it is a connected semisimple Lie group. Theorem 7.1 extends this to every connected locally compact group. Compare this with Theorem 5.5 for discrete groups.

8. Exercises

Exercise 8.1. Let \(M\subset B(H)\) be a von Neumann algebra and let \(\mathcal G\) be a group of automorphisms of \(M\) that is amenable as a discrete group. Show that if \(M\) is injective, then the fixed-point algebra \(M^{\mathcal G}=\{x\in M:\gamma(x)=x\ \text{for all}\ \gamma\in\mathcal G\}\) is injective.

Solution. Let \(m\) be a left-invariant mean on \(\ell^\infty(\mathcal G)\). For \(y\in M\) define \(P(y)\) by \(\langle P(y)\xi,\eta\rangle=m(\gamma\mapsto\langle\gamma(y)\xi,\eta\rangle)\). As in the proof of Theorem 5.1, \(P(y)\) is a well-defined operator with \(\|P(y)\|\le\|y\|\), \(P\) is ucp, and \(P(y)=y\) for \(y\in M^{\mathcal G}\). To see \(P(y)\in M\): if \(a'\in M'\) then \(\langle P(y)a'\xi,\eta\rangle=m(\langle\gamma(y)\xi,a'^*\eta\rangle) =\langle a'P(y)\xi,\eta\rangle\), since \(\gamma(y)\in M\) commutes with \(a'\); so \(P(y)\in M''=M\). Invariance: for \(\beta\in\mathcal G\), \(\beta\) is a normal automorphism (every \(\ast\)-automorphism of a von Neumann algebra is normal), and we must show \(\beta(P(y))=P(y)\). For a normal functional \(\omega\) on \(M\), the map \(\omega\mapsto m(\gamma\mapsto\omega(\gamma(y)))\) is linear and bounded, and it agrees with \(\omega(P(y))\) for vector functionals, hence (by linearity and norm continuity, since normal functionals are norm limits of sums of vector functionals) for all normal \(\omega\). Then \(\omega(\beta(P(y)))=(\omega\circ\beta)(P(y))=m(\gamma\mapsto\omega(\beta\gamma(y)))=m(\gamma\mapsto\omega(\gamma(y))) =\omega(P(y))\) by left invariance, for every normal \(\omega\), so \(\beta(P(y))=P(y)\). Thus \(P(y)\in M^{\mathcal G}\). If \(E\colon B(H)\to M\) is a ucp projection, \(P\circ E\) is a ucp projection onto \(M^{\mathcal G}\), which is therefore injective (it is a von Neumann algebra: it is a unital \(\ast\)-subalgebra, and σ-weakly closed because each \(\gamma\) is normal).

Exercise 8.2 (projections of norm one need not be normal). Let \(A=L^\infty[0,1]\) act on \(L^2[0,1]\) by multiplication. Show that there is a projection of norm one from \(B(L^2[0,1])\) onto \(A\), but no normal one.

Solution. \(A\) is abelian, hence injective (Corollary 5.2(a)), so a projection of norm one exists (Theorem 2.1). Suppose \(E\) is a normal one; it is ucp and \(A\)-bimodular. Let \((e_j)\) be an orthonormal basis of \(L^2[0,1]\) and \(p_j\) the projection onto \(\mathbb Ce_j\). Fix \(j\) and let \(f=E(p_j)\in A\), \(f\ge0\). For a Borel set \(S\) let \(q=1_S\in A\). The vector \(qe_j\) lies in the range of \(q\), so \(qp_jq=|qe_j\rangle\langle qe_j|\le\|qe_j\|^2q\), and bimodularity gives \[ 1_Sf=qE(p_j)q=E(qp_jq)\le\Bigl(\int_S|e_j|^2\Bigr)1_S . \] If \(f>c>0\) on a set \(C\) of positive measure, choose \(S\subset C\) of positive measure so small that \(\int_S|e_j|^2<c\) (possible by absolute continuity of the integral and because Lebesgue measure has no atoms); then \(c<f\le\int_S|e_j|^2<c\) on \(S\), a contradiction. So \(E(p_j)=0\) for every \(j\). By normality, \(1=E(1)=E(\sum_jp_j)=\sum_jE(p_j)=0\), a contradiction.

Exercise 8.3. Show that a von Neumann algebra \(M\) is injective if and only if \(M_n(M)\) is injective for some (equivalently, every) \(n\ge1\).

Solution. \(M_n(M)\cong M\bar\otimes B(\mathbb C^n)\). If \(M\) is injective, so is \(M\bar\otimes B(\mathbb C^n)\) by Proposition 2.3(c). Conversely, \(M\) is isomorphic to the corner \(e_{11}M_n(M)e_{11}\), which is injective by Proposition 2.3(a).

Exercise 8.4. Let \(G\) be a compact group with normalized Haar measure and \(\pi\) a strongly continuous unitary representation on \(H\). Show directly that \(E(y)\), defined by \(\langle E(y)\xi,\eta\rangle=\int_G\langle\pi(g)y\pi(g)^*\xi,\eta\rangle\,dg\), is a ucp projection of \(B(H)\) onto \(\pi(G)'\), and that \(E\) is normal. Compare with Exercise 8.2.

Solution. Haar measure on a compact group is a left-invariant mean on \(C_b(G)=C(G)\), so the proof of Theorem 5.1 with \(N=\mathbb C1\) (hence \(N'=B(H)\), \(E_0=\mathrm{id}\)) shows that \(E\) is a ucp projection onto \(\pi(G)'\). Normality: if \(y_i\uparrow y\) is a bounded increasing net in \(B(H)_+\), then for each \(\xi\) the continuous functions \(g\mapsto\langle\pi(g)y_i\pi(g)^*\xi,\xi\rangle\) increase to the continuous function \(g\mapsto\langle\pi(g)y\pi(g)^*\xi,\xi\rangle\); by Dini's theorem the convergence is uniform on the compact group, so the integrals converge and \(E(y_i)\uparrow E(y)\). The contrast with Exercise 8.2: there the projection comes from averaging over the unitary group of \(L^\infty[0,1]\) as a discrete group, an infinite group whose invariant means are only finitely additive, and normality is lost.

Exercise 8.5. Let \(b\in B(H)_{sa}\) and \(M=\mathbb C1\subset B(H)\). Show that the element \(x=\omega(b)1\), for any state \(\omega\) on \(B(H)\), solves every \(b\)-constraint, and explain why this is consistent with Theorem 3.4.

Solution. Here \(s\in M_n(\mathbb C1)_{sa}\) is \(1\otimes t\) with \(t\in(M_n)_{sa}\), and the constraint reads \(b\otimes\sigma\le1\otimes t\). The map \(\omega_n=\omega\otimes\mathrm{id}\colon M_n(B(H))\to M_n\) is positive (a state is completely positive), so \(\omega(b)\sigma\le t\), that is \(\omega(b)1\otimes\sigma\le s\), and \(|\omega(b)|\le\|b\|\). This is consistent with Theorem 3.4 because \(\mathbb C1\) is injective (Example 2.4), and the solution found is exactly \(E(b)\) for the projection \(E(y)=\omega(y)1\) used in the proof of (i)⇒(ii).

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