Arch. Math. 97 (2011), 151–156c© 2011 Springer Basel AG0003-889X/11/020151-6published online July 16, 2011DOI 10.1007/s00013-011-0287-5 Archiv der Mathematik

On weakly injective von Neumann algebras

Florin Pop

Abstract. A von Neumann algebra M ⊂ B(H) is called weakly injectiveif there exist an ultraweakly dense unital C∗-subalgebra A ⊂ M and aunital completely positive map ϕ : B(H) → M such that ϕ(a) = a for alla ∈ A. In this note we present several properties of weakly injective vonNeumann algebras and highlight the role these algebras play in relationto the QWEP conjecture.

Mathematics Subject Classification (2000). 46L05, 46L06, 46L10.

Keywords. C∗-algebra, von Neumann algebra, Weakly injective,QWEP conjecture.

1. Introduction and preliminary results. A C∗-algebra A has the Weak Expec-tation Property (WEP, [6]) if it can be faithfully represented on a Hilbertspace H such that there exists a u.c.p. (unital completely positive) map ϕ :B(H) → A∗∗ satisfying ϕ(a) = a for all a ∈ A. This definition does not dependon the particular representation of A.

A has the Lifting Property (LP) if for every u.c.p. map ϕ : A → B/J thereexists a u.c.p. map ψ : A → B such that ϕ = π ◦ ψ, where π is the quotientmap π : B → B/J . A has the Local Lifting Property (LLP) if, in the abovedefinition, for any finite dimensional operator system E ⊂ A, there exists au.c.p. map ψ : E → B such that ϕ = π ◦ ψ on E.

The prototype examples are B(H) for the WEP and C∗(F∞) (the fullC∗-algebra of the free group on countably many generators) for the LLP. Wenote that C∗(F∞) actually has the stronger LP ([5]).

In his groundbreaking paper [4], Kirchberg highlighted new directions ofinquiry in the theory of C∗-algebras. He proved several deep results whichconnect tensor products, the WEP and LLP, as well as Connes’ embeddingproblem. We recall some of these results in the following

152 F. Pop Arch. Math.

Proposition 1.1. (1) If A has the WEP and B has the LLP, then A⊗minB =A⊗max B.

(2) A has the LLP if and only if A⊗min B(H) = A⊗max B(H).(3) A has the WEP if and only if A⊗min C

∗(F∞) = A⊗max C∗(F∞).

(4) The following are equivalent:(i) All separable C∗-algebras are QWEP (that is, quotients of

C∗-algebras with the WEP).(ii) C∗(F∞) ⊗min C

∗(F∞) = C∗(F∞) ⊗max C∗(F∞).

(iii) Every type II1 von Neumann algebra with separable predual is*-isomorphic to a subalgebra of Rω.

Recall that, if R is the hyperfinite type II1 factor with trace τ and ωis a free ultrafilter on N, then Rω denotes the quotient (ΠnR)/Nω, whereNω = {(an)n≥1 ∈ ΠnR : limω τ(a∗

nan) = 0}. The equivalent properties (i) and(iii) in Proposition 1.1(4) are known as the QWEP conjecture and, respectively,Connes’ embedding problem.

We should also point out that one of the main questions in [4], namelythe existence of a unique C∗-norm on B(H) ⊗ B(H), was answered in thenegative in [3]. In particular, B(H) has the WEP but not the LLP. The equiv-alent statements in Proposition 1.1(4) are currently open problems and thereis growing interest and effort in their study.

We continue this section with a few well-known technical facts which willbe used subsequently. The first result is a consequence of the fact that com-pletely positive maps are modular on their multiplicative domains. We referthe reader to Section 3.6 of [2] for more information.

Lemma 1.2. Let ϕ : A⊗minB → B(H) be a u.c.p. map whose restriction to Bis a *-homomorphism. Then ϕ(A) ⊂ ϕ(B)′.

Proof. Denote the restriction of ϕ to B by π. We use Corollary 2.8 in [7] toobtain, for all a ∈ A and b ∈ B, that ϕ(a)ϕ(b) = ϕ(a)π(b) = ϕ(ab) = ϕ(ba) =π(b)ϕ(a) = ϕ(b)ϕ(a). �

Next, we collect several properties of C∗-algebras with the WEP.

Proposition 1.3. (1) Let A ⊂ B(H) be a C∗-algebra with the WEP. Thenthere exists a u.c.p. map ϕ : B(H) → A′′ such that ϕ(a) = a for everya ∈ A.

(2) A von Neumann algebra has the WEP if and only if it is injective.(3) If G is a countable, discrete group, then G is amenable if and only if

C∗red(G) has the WEP.

Proof. (1) follows from the definition of the WEP combined with the existenceof a conditional expectation from A∗∗ onto A′′. (2) follows from (1), while (3)follows from an argument contained in the proof of Proposition 4.2 in [6]. �

We conclude this section with a well-known embedding result.

Lemma 1.4. Let H and M ⊂ B(H) be a separable Hilbert space and a vonNeumann algebra without a type I finite direct summand, respectively. Thenthere exists a completely isometric u.c.p. map from B(H) to M.

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Proof. Let (en)n≥1 be a sequence of mutually orthogonal projections inM,

∑ei = I. Since M has no direct summand of type I, the algebras

enMen contain matrices of any order. Let (pn)n≥1 be an increasing sequenceof projections in B(H),dim pnH = n. Since for every T ∈ B(H) one has||T || = supn ||pnTpn||, it follows that the map T → ⊕

n pnTpn establishesa completely isometric, u.c.p. embedding B(H) → ΠnMn. Then B(H) ⊂ΠnMn ⊂ ΠnenMen ⊂ M, where the inclusions are completely isometric, u.c.p.embeddings. �2. Weakly injective von Neumann algebras. This section contains the mainresults and is largely motivated by Brown’s work [1] and by Proposition 1.3.Von Neumann algebras M containing an ultraweakly (i.e., σ(M,M∗)) denseC∗-subalgebra A with the property that there exists a u.c.p. map from B(H)to M acting identically on A can be traced back to [6], where A was said tohave a weak expectation. In [1] this property was referred to as M havingthe WEP relative to A. Brown [1] proved that type II1 factors with separablepredual have this property if and only if they are embeddable in Rω. Due tothe importance of this circle of ideas, we propose the following

Definition 2.1. Let M ⊂ B(H) be a von Neumann algebra. M is called weaklyinjective if there exist an ultraweakly dense unital C∗-subalgebra A ⊂ M anda u.c.p. map ϕ : B(H) → M such that ϕ(a) = a for all a ∈ A.

Obviously injectivity implies weak injectivity, so we can restrict attentionto type II and III von Neumann algebras. Here we have the advantage thatthese algebras contain ultraweakly dense copies of C∗(F∞) ([1]). The samecan be said in the type I∞ case, but these algebras are already known to beinjective. We must point out that, while the statement of Proposition 3.1 in[1] is proved for type II1 factors only, the argument works word for word inthe type II and III cases, provided the predual is separable. The main reasonis that these algebras, as well as any of their corners cut by projections, con-tain matrix subalgebras of any order. For the remainder of this section, allvon Neumann algebras (except, of course, Rω) are assumed to have separablepredual.

Proposition 2.2. If M ⊂ B(H) is a QWEP von Neumann algebra, then M isweakly injective.

Proof. The direct summand of type I is injective, while the summands of typeII and III contain ultraweakly dense copies of C∗(F∞) ([1]). Corollary 3.8 in[4] shows that the latter direct summands are weakly injective (relative to thecopies of C∗(F∞)). Since direct sums of weakly injective algebras are easilyseen to be weakly injective, the conclusion follows. �

A positive answer to the QWEP conjecture would therefore imply that allseparable von Neumann algebras are weakly injective (Proposition 2.8 belowshows that the two statements are actually equivalent). In the type II1 case,weak injectivity is equivalent to Connes’ embedding problem. This makes thetype III case, where very little is presently known, of particular interest: areall type III von Neumann algebras weakly injective?

154 F. Pop Arch. Math.

We continue this section with two basic properties: the first ensures thatweak injectivity does not depend on the Hilbert space representation, whilethe second one shows that weak injectivity passes to the commutant. If A,B ⊂B(H) are C∗-algebras, we will denote by C∗(A,B) the C∗-algebra generatedby A and B.

Proposition 2.3. Let M1 ⊂ B(H) and M2 ⊂ B(K) be *-isomorphic vonNeumann algebras. If M1 is weakly injective, then so is M2.

Proof. Let θ : M1 → M2 a *-isomorphism and let A ⊂ M1 be an ultraweaklydense C∗-subalgebra together with a u.c.p. map ϕ : B(H) → M1 such thatϕ(a) = a for all a ∈ A. If B = θ(A), then B is ultraweakly dense in M2 becauseθ is normal. The *-homomorphism θ−1 extends to a u.c.p. map Φ : B(K) →B(H). The composite map θ ◦ϕ ◦ Φ is a u.c.p. map from B(K) with values inM2 satisfying, for every b = θ(a), (θ ◦ ϕ ◦ Φ)(b) = (θ ◦ ϕ)(a) = θ(a) = b. �

Proposition 2.4. If a von Neumann algebra M ⊂ B(H) is weakly injective,then so is M ′.

Proof. Let A ⊂ M be an ultraweakly dense C∗-subalgebra together with au.c.p. map ϕ : B(H) → M such that ϕ(a) = a for all a ∈ A. We mayassume that M (and hence M ′) is of type II or III, so let B ⊂ M ′ be anultraweakly dense C∗-subalgebra isomorphic to C∗(F∞). By composing the*-homomorphism from M ⊗max B with values in C∗(M,B) sending m ⊗ bto mb with the u.c.p. map ϕ ⊗ id : B(H) ⊗max B → M ⊗max B (see [7]),we obtain a u.c.p. map from B(H) ⊗max B to C∗(M,B) which sends x ⊗ bto ϕ(x)b. Since B(H) ⊗max B = B(H) ⊗min B, this u.c.p. map extends toB(H) ⊗min B(H). This extension is the identity on A, so by Lemma 1.2,the restriction to B(H) = I ⊗ B(H) takes values in M ′ and acts identicallyon B. �

The next result and the remarks following it emphasize the role played inweak injectivity by subalgebras isomorphic to C∗(F∞).

Proposition 2.5. Let M ⊂ B(H) be a von Neumann algebra with a C∗-subalgebra A ⊂ M isomorphic to C∗(F∞). If N is a weakly injective vonNeumann algebra, then every u.c.p. map ϕ : A → N extends to a u.c.p. mapΦ : M → N.

Proof. By Proposition 2.4, N ′ is weakly injective, so choose an ultraweaklydense C∗-subalgebra B ⊂ N ′ and a u.c.p. map ψ : B(H) → N ′ such thatψ(b) = b for all b ∈ B. Like in the proof of Proposition 2.4, we obtain au.c.p. map from N ⊗maxB(H) to C∗(N,N ′) which sends n⊗x to nψ(x). Theu.c.p. map ϕ : A → N extends to a u.c.p. map ϕ ⊗ id from A ⊗max B(H)to N ⊗max B(H). By composing the previous two maps, we obtain a u.c.p.map from A ⊗max B(H) to C∗(N,N ′) which sends a ⊗ x to ϕ(a)ψ(x). SinceA⊗maxB(H) = A⊗minB(H), this u.c.p. map extends to B(H)⊗minB(H). Thisextension is the identity on B, so by Lemma 1.2, the restriction to B(H) =B(H) ⊗ I takes values in B′ = N and its restriction to A is ϕ. �

Vol. 97 (2011) Weakly injective algebras 155

Corollary 2.6. (i) If M ⊂ B(H) is weakly injective and of type II or III, thenthe ultraweakly dense subalgebra A ⊂ M in Definition 2.1 can be chosen to beisomorphic to C∗(F∞).

(ii) If M ⊂ B(H) is weakly injective with a C∗-subalgebra A ⊂ M iso-morphic to C∗(F∞), then every u.c.p. map ϕ : A → M (in particular everyautomorphism of A) extends to a u.c.p. map Φ : M → M.

Another consequence of Proposition 2.5 pertains to local approximationsby operator systems. In [1] Brown proved the surprising fact that if M ⊂ Rω

is a subfactor, F ⊂ M is a finite set and ε > 0 is fixed, then there existsa complete order embedding θ : R → M (i.e., θ is completely positive andθ−1 : θ(R) → R is also completely positive) such that for each x ∈ F thereexists r ∈ R such that ||x−θ(r)||2 < ε. It turns out that subfactors of Rω havea universal property of this kind: they are, locally, completely order isomor-phic to any von Neumann algebra rich enough to contain a copy of C∗(F∞),in particular to any type II or III von Neumann algebra.

Proposition 2.7. Let M ⊂ B(H) be a von Neumann algebra with aC∗-subalgebra A ⊂ M isomorphic to C∗(F∞). If N ⊂ Rω is a subfactor,F ⊂ N is a finite set and ε > 0 is fixed, then there exists a complete orderembedding θ : M → N such that for each x ∈ F there exists m ∈ M such that||x− θ(m)||2 < ε.

Proof. We may assume without loss of generality that ||x|| ≤ 1 for all x ∈ F.Let p ∈ N, p �= I be a projection of trace τ(p) > 1 − δ, where δ < ε2 issmall enough to ensure ||xi − pxip||2 < ε, i = 1, 2, . . . , n. Since pNp is also asubfactor of Rω, it is weakly injective. Let B ⊂ pNp denote an ultraweaklydense copy of C∗(F∞) and let π : A → B be a *-isomorphism. By Proposi-tion 2.5, there exists a u.c.p. map ϕ : M → pNp extending π. On the otherhand, Lemma 1.4 ensures the existence of a completely isometric u.c.p. mapν : M → (I−p)N(I−p). We claim that the desired complete order embeddingis ϕ ⊕ ν. Let bi ∈ B be such that ||bi − pxip||2 ≤ ε. By Kaplanski’s densitytheorem we may assume that ||bi|| ≤ 1. If ϕ(ai) = bi, then ||ai|| ≤ 1 and wehave

||ϕ⊕ ν(ai) − xi||2 ≤ ||ϕ(ai) − pxip||2 + ||pxip− xi||2 + ||ν(ai)||2 ≤ 3ε

�

We conclude with a result that places weak injectivity in the context ofKirchberg’s equivalent conjectures.

Proposition 2.8. The following are equivalent:(i) C∗(F∞) ⊗min C

∗(F∞) = C∗(F∞) ⊗max C∗(F∞).

(ii) All separable C∗-algebras are QWEP.(iii) All von Neumann algebras with separable predual are weakly injective.

Proof. If (i) holds, then C∗(F∞) has the WEP, so (ii) follows, since every sep-arable C∗-algebra is a quotient of C∗(F∞). If (ii) holds, then by Proposition3.3(v) in [4], all von Neumann algebras (with separable predual) are QWEP.

156 F. Pop Arch. Math.

Then (iii) follows from Proposition 2.2. To prove (iii)⇒(i), we must show thatthe kernel of the canonical *-homomorphism θ : C∗(F∞) ⊗max C

∗(F∞) →C∗(F∞) ⊗min C

∗(F∞) is trivial. Let π : C∗(F∞) ⊗max C∗(F∞) → B(H) be a

faithful representation. The restrictions π1 : A = C∗(F∞) ⊗ I → B(H) andπ2 : B = I ⊗ C∗(F∞) → B(H) are faithful representations with commutingranges such that π(a ⊗ b) = π1(a)π2(b) for all a ∈ A and b ∈ B. Assumingthat π1(A)′′ is weakly injective, there exists, by Corollary 2.6, a u.c.p. mapϕ : B(H) → π1(A)′′ such that ϕ(π1(a)) = π1(a) for all a ∈ A. Since π1(A)′′

and π2(B) commute, there exists a *-homomorphism α : π1(A)′′⊗maxπ2(B) →C∗(π1(A)′′, π2(B)) taking x⊗ y to xy. We can see now that π is equal to thecomposition of the sequence of maps

A⊗max Bθ−→A⊗min B

π1⊗id−→ π1(A) ⊗min B ↪→B(H) ⊗min B=B(H) ⊗max B

id⊗π2−→ B(H) ⊗max π2(B)ϕ⊗id−→ π1(A)′′ ⊗max π2(B) α−→ C∗(π1(A)′′, π2(B))

therefore ker θ ⊂ ker π. But π is faithful, so ker θ = 0, which concludes theproof. �Acknowledgements. I would like to thank Professor Nathanial Brown for use-ful discussions.

References

[1] N. P. Brown, Connes’ embedding problem and Lance’s WEP, Int. Math. Res.

Not. 10 (2004), 501–510.

[2] N. P. Brown and N. Ozawa, C∗-algebras and finite-dimensional approxima-

tions, Graduate Studies in Mathematics 88, American Mathematical Society,

Providence, RI, 2008.

[3] M. Junge and G. Pisier, Bilinear forms on exact operator spaces and B(H) ⊗B(H), Geom. Funct. Anal. 5 (1995), 329–363.

[4] E. Kirchberg, On nonsemisplit extensions, tensor products and exactness of

group C∗-algebras, Invent. Math. 112 (1993), 449–489.

[5] E. Kirchberg, Commutants of unitaries in UHF algebras and functorial prop-

erties of exactness, J. Reine Angew. Math. 452 (1994), 39–77.

[6] E. C. Lance, On nuclear C∗-algebras, J. Funct. Anal. 12 (1973), 157–176.

[7] N. Ozawa, About the QWEP conjecture, Int. J. Math. 15 (2004), 501–530.

Florin PopDepartment of Mathematics and Computer Science,Wagner College,Staten Island, New York 10301,USAe-mail: fpop@wagner.edu

Received: 4 November 2010