Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar echal - - PowerPoint PPT Presentation

. . . . . .

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Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology

Hiroshi ANDO

Erwin Schr¨

• dinger Institute, Vienna

ENS Lyon, 27.9.2013 Joint work with Uffe Haagerup and Carl Winsløw (University of Copenhagen)

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 1 / 24

. . . . . .

Outline of Talk

. .

1

Kirchberg’s QWEP Conjecture . .

2

Effros-Mar´ echal Topology . .

3

Ultraproduct of von Neumann algebras . .

4

Characterizations of QWEP von Neumann Algebras

• H. Ando, U. Haagerup, “Ultraproucts of von Neumann algebras”,

arXiv:1212.5457

• H. Ando, U. Haagerup, C. Winsløw, “Ultraproducts, QWEP von

Neumann algebras, and the Effros-Mar´ echal topology”, arXiv:1306.0460

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 2 / 24

. . . . . .

QWEP Conjecture

Kirchberg (’93) revealed remarkable connetions among Tensor products of C∗-algebras Lance’s Weak Expectation Property (WEP) Connes’s Embedding Conjecture (CEC): ∀N sep. II1 factor embeds into Rω?

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 3 / 24

. . . . . .

QWEP Conjecture

Kirchberg (’93) revealed remarkable connetions among Tensor products of C∗-algebras Lance’s Weak Expectation Property (WEP) Connes’s Embedding Conjecture (CEC): ∀N sep. II1 factor embeds into Rω? In this talk, we discuss how QWEP property is connected to ultraproducts

• f von Neumann algebras using topological method.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 3 / 24

. . . . . .

QWEP Conjecture

.

Definition (Lance ’73, Kirchberg ’93)

. . (1) C∗-alg A has the weak expectation property (WEP) if for any faithful representation A ⊂ B(H), there is a ucp map Φ: B(H) → A∗∗ s.t. Φ|A = idA. (2) C∗-alg A has the quotient weak expectation property (QWEP) if it is the quotient of a C∗-algebra with WEP. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 4 / 24

. . . . . .

QWEP Conjecture

.

Definition (Lance ’73, Kirchberg ’93)

. . (1) C∗-alg A has the weak expectation property (WEP) if for any faithful representation A ⊂ B(H), there is a ucp map Φ: B(H) → A∗∗ s.t. Φ|A = idA. (2) C∗-alg A has the quotient weak expectation property (QWEP) if it is the quotient of a C∗-algebra with WEP. .

Theorem (Kirchberg’s QWEP Conjecture)

. . TFAE. (1) C∗(F∞) ⊗min C∗(F∞) = C∗(F∞) ⊗max C∗(F∞). (2) Every C∗-algebra has QWEP. (3) C∗(F∞) has WEP. (4) (Connes’s Embedding Conjecture) Every separable type II1 factor M admits an embedding into Rω, where R is the hyperfinite II1 factor.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 4 / 24

. . . . . .

.

Theorem (Kirchberg ’93)

. . A separable II1 factor M embeds into Rω if and only if M has QWEP. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 5 / 24

. . . . . .

.

Theorem (Kirchberg ’93)

. . A separable II1 factor M embeds into Rω if and only if M has QWEP. Is QWEP conjecture true? C∗(F∞) ⊗min C∗(F∞)

?

= C∗(F∞) ⊗max C∗(F∞) Kirchberg(’93) proved C∗(F∞) ⊗min B(ℓ2) = C∗(F∞) ⊗max B(ℓ2). . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 5 / 24

. . . . . .

.

Theorem (Kirchberg ’93)

. . A separable II1 factor M embeds into Rω if and only if M has QWEP. Is QWEP conjecture true? C∗(F∞) ⊗min C∗(F∞)

?

= C∗(F∞) ⊗max C∗(F∞) Kirchberg(’93) proved C∗(F∞) ⊗min B(ℓ2) = C∗(F∞) ⊗max B(ℓ2). .

Theorem (Junge-Pisier ’95)

. . B(ℓ2) ⊗min B(ℓ2) ̸= B(ℓ2) ⊗max B(ℓ2).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 5 / 24

. . . . . .

Effros-Mar´ echal Topology

Fix H ∼ = ℓ2. vN(H)=set of all vNas on H. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 6 / 24

. . . . . .

Effros-Mar´ echal Topology

Fix H ∼ = ℓ2. vN(H)=set of all vNas on H. Effros (’65) introduced Effros Borel structure on vN(H). Mar´ echal (’73) introduced Polish topology on vN(H) that generates Effros Borel structure. Haagerup-Winsløw (’98,’00) studied the Effros-Mar´ echal topology. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 6 / 24

. . . . . .

Effros-Mar´ echal Topology

Fix H ∼ = ℓ2. vN(H)=set of all vNas on H. Effros (’65) introduced Effros Borel structure on vN(H). Mar´ echal (’73) introduced Polish topology on vN(H) that generates Effros Borel structure. Haagerup-Winsløw (’98,’00) studied the Effros-Mar´ echal topology. .

Definition (Mar´ echal ’73)

. . The Effros-Mar´ echal Topology on vN(H) is the weakest topology which makes all the maps of the form vN(H) ∋ M → ∥φ|M∥, φ ∈ B(H)∗ continuous.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 6 / 24

. . . . . .

.

Definition (Haagerup-Winsløw ’98)

. . For {Mn}∞

n=1 ⊂ vN(H), define lim sup n→∞ Mn and lim inf n→∞ Mn by

(1) lim inf

n→∞ Mn={x ∈ B(H); xn so∗

→ x, ∃(xn)n ∈ ℓ∞(N, Mn)}. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 7 / 24

. . . . . .

.

Definition (Haagerup-Winsløw ’98)

. . For {Mn}∞

n=1 ⊂ vN(H), define lim sup n→∞ Mn and lim inf n→∞ Mn by

(1) lim inf

n→∞ Mn={x ∈ B(H); xn so∗

→ x, ∃(xn)n ∈ ℓ∞(N, Mn)}. (2) lim sup

n→∞ Mn=vNa generated by {x ∈ B(H); x is a weak-limit

point of ∃(xn)n ∈ ℓ∞(N, Mn)}. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 7 / 24

. . . . . .

.

Definition (Haagerup-Winsløw ’98)

. . For {Mn}∞

n=1 ⊂ vN(H), define lim sup n→∞ Mn and lim inf n→∞ Mn by

(1) lim inf

n→∞ Mn={x ∈ B(H); xn so∗

→ x, ∃(xn)n ∈ ℓ∞(N, Mn)}. (2) lim sup

n→∞ Mn=vNa generated by {x ∈ B(H); x is a weak-limit

point of ∃(xn)n ∈ ℓ∞(N, Mn)}. .

Theorem (Haagerup-Winsløw ’98)

. . TFAE. (1) Mn → M in vN(H). (2) lim inf

n→∞ Mn = M = lim sup n→∞ Mn.

Moreover, ( lim sup

n→∞ Mn

)′ = lim inf

n→∞ M ′ n holds.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 7 / 24

. . . . . .

Important subsets of vN(H): F: factors, Finj injective factors, vN(H)st standardly acting vNas, FII1 type II1 factors, etc. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 8 / 24

. . . . . .

Important subsets of vN(H): F: factors, Finj injective factors, vN(H)st standardly acting vNas, FII1 type II1 factors, etc. .

Theorem (Haagerup-Winsløw ’00)

. . Subset of vN(H) Dense in vN(H)? Gδ? F Yes Yes ∪

n≤n0 FIn, n0 ∈ N

No Yes (closed) FIfin * No (but Fσ) FI∞ * No (but Fσ) FII1 Yes No FII∞ Yes No FIII0 Yes No FIIIλ, λ ∈ (0, 1) Yes No FIII1 Yes Yes Finj * Yes Fst Yes Yes

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 8 / 24

. . . . . .

Important subsets of vN(H): F: factors, Finj injective factors, vN(H)st standardly acting vNas, FII1 type II1 factors, etc. .

Theorem (Haagerup-Winsløw ’00)

. . Subset of vN(H) Dense in vN(H)? Gδ? F Yes Yes ∪

n≤n0 FIn, n0 ∈ N

No Yes (closed) FIfin * No (but Fσ) FI∞ * No (but Fσ) FII1 Yes No FII∞ Yes No FIII0 Yes No FIIIλ, λ ∈ (0, 1) Yes No FIII1 Yes Yes Finj * Yes Fst Yes Yes Moreover, * are all equivalent to QWEP (Connes Embedding) conjecture.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 8 / 24

. . . . . .

Quick reminder: .

Definition

. . (M, H, J, P♮

M) is called a standard form of M if J : H → H is an

antilinear involution, P♮

M = (P♮ M)0 self-dual convex cone in H such that

(1) JMJ = M ′. (2) Jξ = ξ, ξ ∈ P♮

M.

(3) xJxJP ⊂ P♮

M,

x ∈ M. (4) JxJ = x∗, x ∈ Z(M).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 9 / 24

. . . . . .

Quick reminder: .

Definition

. . (M, H, J, P♮

M) is called a standard form of M if J : H → H is an

antilinear involution, P♮

M = (P♮ M)0 self-dual convex cone in H such that

(1) JMJ = M ′. (2) Jξ = ξ, ξ ∈ P♮

M.

(3) xJxJP ⊂ P♮

M,

x ∈ M. (4) JxJ = x∗, x ∈ Z(M). GNS rep (πφ, Hφ, ξφ) of f.n. state φ gives a standard form. In this case, P♮

M = ∆

1 4

φM+ξφ, J = Jφ.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 9 / 24

. . . . . .

Quick reminder: .

Definition

. . (M, H, J, P♮

M) is called a standard form of M if J : H → H is an

antilinear involution, P♮

M = (P♮ M)0 self-dual convex cone in H such that

(1) JMJ = M ′. (2) Jξ = ξ, ξ ∈ P♮

M.

(3) xJxJP ⊂ P♮

M,

x ∈ M. (4) JxJ = x∗, x ∈ Z(M). GNS rep (πφ, Hφ, ξφ) of f.n. state φ gives a standard form. In this case, P♮

M = ∆

1 4

φM+ξφ, J = Jφ.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 9 / 24

. . . . . .

Quick reminder: .

Definition

. . (M, H, J, P♮

M) is called a standard form of M if J : H → H is an

antilinear involution, P♮

M = (P♮ M)0 self-dual convex cone in H such that

(1) JMJ = M ′. (2) Jξ = ξ, ξ ∈ P♮

M.

(3) xJxJP ⊂ P♮

M,

x ∈ M. (4) JxJ = x∗, x ∈ Z(M). GNS rep (πφ, Hφ, ξφ) of f.n. state φ gives a standard form. In this case, P♮

M = ∆

1 4

φM+ξφ, J = Jφ.∀M, ∃! (M, J, P, H)

(Haagerup ’75).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 9 / 24

. . . . . .

Very useful tricks: .

Theorem (Expectation-Trick, Haagerup-Winsløw ’00)

. . Assume N ⊂ M be ∞-dim vNas on H, N standard on H, and there is faithful normal expectation E : M → N. Then ∃un ∈ U(H) and ∃M0 ∼ = M s.t. unM0u∗

n → N in vN(H).

. . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 10 / 24

. . . . . .

Very useful tricks: .

Theorem (Expectation-Trick, Haagerup-Winsløw ’00)

. . Assume N ⊂ M be ∞-dim vNas on H, N standard on H, and there is faithful normal expectation E : M → N. Then ∃un ∈ U(H) and ∃M0 ∼ = M s.t. unM0u∗

n → N in vN(H).

.

Theorem (⊗-trick, Haagerup-Winsløw’00)

. . Let K ∼ = ℓ2, and v0 ∈ U(H ⊗ K, H). Then ∃un ∈ U(H ⊗ K) s.t. for any N ∈ vN(H) and M ∈ vN(K), one has v0u∗

n(N⊗M)u0v∗ n→∞

→ N in vN(H).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 10 / 24

. . . . . .

Very useful tricks: .

Theorem (Expectation-Trick, Haagerup-Winsløw ’00)

. . Assume N ⊂ M be ∞-dim vNas on H, N standard on H, and there is faithful normal expectation E : M → N. Then ∃un ∈ U(H) and ∃M0 ∼ = M s.t. unM0u∗

n → N in vN(H).

.

Theorem (⊗-trick, Haagerup-Winsløw’00)

. . Let K ∼ = ℓ2, and v0 ∈ U(H ⊗ K, H). Then ∃un ∈ U(H ⊗ K) s.t. for any N ∈ vN(H) and M ∈ vN(K), one has v0u∗

n(N⊗M)u0v∗ n→∞

→ N in vN(H). ⇝ Type III factors are dense, McDuff factors are dense in factors F, etc...

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 10 / 24

. . . . . .

Effros-Mar´ echal Topology

.

Theorem (Haagerup-Winsløw ’00)

. . Let M be a II1 factor on H. TFAE. (1) M ∈ Finj. (2) There is an embedding i : M → Rω.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 11 / 24

. . . . . .

Effros-Mar´ echal Topology

.

Theorem (Haagerup-Winsløw ’00)

. . Let M be a II1 factor on H. TFAE. (1) M ∈ Finj. (2) There is an embedding i : M → Rω. Goal of today’s talk: further investigation of Finj and study the connection of the following: Effros-Mar´ echal

• QWEP vNas

Ultraproducts

• Hiroshi ANDO (Erwin Schr¨
• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 11 / 24

. . . . . .

The Ocneanu Ultaproduct

(Mn, φn)∞

n=1 sequence of vNas/n.f.states. ω ∈ βN \ N.

.

Definition (Ocneanu ’85)

. . Define Ocneanu ultraproduct (Mn, φn)ω := Mω(Mn, φn)/Iω(Mn, φn), where Iω(Mn, φn) := {(xn)n ∈ ℓ∞(N, Mn); ∥xn∥♯

φn n→ω

→ 0}, Mω(Mn, φn) := {x ∈ ℓ∞(N, Mn); xIω + Iωx ⊂ Iω}. φω : (xn)ω → limω φn(xn) is a n.f. state on (Mn, φn)ω. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 12 / 24

. . . . . .

The Ocneanu Ultaproduct

(Mn, φn)∞

n=1 sequence of vNas/n.f.states. ω ∈ βN \ N.

.

Definition (Ocneanu ’85)

. . Define Ocneanu ultraproduct (Mn, φn)ω := Mω(Mn, φn)/Iω(Mn, φn), where Iω(Mn, φn) := {(xn)n ∈ ℓ∞(N, Mn); ∥xn∥♯

φn n→ω

→ 0}, Mω(Mn, φn) := {x ∈ ℓ∞(N, Mn); xIω + Iωx ⊂ Iω}. φω : (xn)ω → limω φn(xn) is a n.f. state on (Mn, φn)ω. .

Remark

. . Ocneanu considered the case Mn ≡ M, φn ≡ φ. In this case (M, φ)ω is independent of φ, so denote it as M ω. However, it is crucial to use the sequence {φn}∞

n=1 to study

Effros-Mar´ echal topology.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 12 / 24

. . . . . .

The Groh-Raynaud Ultraproduct

Taking Ocneanu UP does not commute with taking NC Lp-spaces: for p = 1, B(ℓ2)ω = B(ℓ2), but (B(ℓ2)∗)ω ̸= B(ℓ2)∗ . . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 13 / 24

. . . . . .

The Groh-Raynaud Ultraproduct

Taking Ocneanu UP does not commute with taking NC Lp-spaces: for p = 1, B(ℓ2)ω = B(ℓ2), but (B(ℓ2)∗)ω ̸= B(ℓ2)∗ Is there a right UP M → M ω s.t. (Lp(M))ω = Lp( M ω)? . . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 13 / 24

. . . . . .

The Groh-Raynaud Ultraproduct

Taking Ocneanu UP does not commute with taking NC Lp-spaces: for p = 1, B(ℓ2)ω = B(ℓ2), but (B(ℓ2)∗)ω ̸= B(ℓ2)∗ Is there a right UP M → M ω s.t. (Lp(M))ω = Lp( M ω)? YES! (Groh ’86). More precisely (Raynaud ’02): . . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 13 / 24

. . . . . .

The Groh-Raynaud Ultraproduct

Taking Ocneanu UP does not commute with taking NC Lp-spaces: for p = 1, B(ℓ2)ω = B(ℓ2), but (B(ℓ2)∗)ω ̸= B(ℓ2)∗ Is there a right UP M → M ω s.t. (Lp(M))ω = Lp( M ω)? YES! (Groh ’86). More precisely (Raynaud ’02): .

Theorem (Raynaud ’02)

. . For each {Mn}∞

n=1, !∃ ∏ω Mn s.t.

Lp(∏ω Mn) = (Lp(Mn))ω (1 ≤ p < ∞). In particular, (∏ω Mn)∗ = ((Mn)∗)ω. (∏ω Mn)′ = ∏ω M ′

n (in standard form)

. . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 13 / 24

. . . . . .

The Groh-Raynaud Ultraproduct

Taking Ocneanu UP does not commute with taking NC Lp-spaces: for p = 1, B(ℓ2)ω = B(ℓ2), but (B(ℓ2)∗)ω ̸= B(ℓ2)∗ Is there a right UP M → M ω s.t. (Lp(M))ω = Lp( M ω)? YES! (Groh ’86). More precisely (Raynaud ’02): .

Theorem (Raynaud ’02)

. . For each {Mn}∞

n=1, !∃ ∏ω Mn s.t.

Lp(∏ω Mn) = (Lp(Mn))ω (1 ≤ p < ∞). In particular, (∏ω Mn)∗ = ((Mn)∗)ω. (∏ω Mn)′ = ∏ω M ′

n (in standard form)

.

Remark

. . M ω and ∏ω M are VERY different: in fact (Raynaud) ∏ω B(ℓ2) is not semifinite.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 13 / 24

. . . . . .

The Groh-Raynaud Ultraproduct

Taking Ocneanu UP does not commute with taking NC Lp-spaces: for p = 1, B(ℓ2)ω = B(ℓ2), but (B(ℓ2)∗)ω ̸= B(ℓ2)∗ Is there a right UP M → M ω s.t. (Lp(M))ω = Lp( M ω)? YES! (Groh ’86). More precisely (Raynaud ’02): .

Theorem (Raynaud ’02)

. . For each {Mn}∞

n=1, !∃ ∏ω Mn s.t.

Lp(∏ω Mn) = (Lp(Mn))ω (1 ≤ p < ∞). In particular, (∏ω Mn)∗ = ((Mn)∗)ω. (∏ω Mn)′ = ∏ω M ′

n (in standard form)

.

Remark

. . M ω and ∏ω M are VERY different: in fact (Raynaud) ∏ω B(ℓ2) is not

• semifinite. Moreover (AH’12), while Rω is a II1 factor, ∏ω R is neither

semifnite nor a factor!

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 13 / 24

. . . . . .

Ocneanu vs Groh-Raynaud

M ω ̸= ∏ω M in many ways. But are they unrelated? Consider {Mn, φn}∞

n=1. Then

φω = (φn)ω ∈ ((Mn)∗)ω = (∏ω Mn)∗. Let p = supp(φω) ∈ ∏ω Mn. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 14 / 24

. . . . . .

Ocneanu vs Groh-Raynaud

M ω ̸= ∏ω M in many ways. But are they unrelated? Consider {Mn, φn}∞

n=1. Then

φω = (φn)ω ∈ ((Mn)∗)ω = (∏ω Mn)∗. Let p = supp(φω) ∈ ∏ω Mn. .

Theorem (A-Haagerup’12)

. . p(∏ω Mn)p ∼ = (Mn, φn)ω holds.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 14 / 24

. . . . . .

Ocneanu vs Groh-Raynaud

M ω ̸= ∏ω M in many ways. But are they unrelated? Consider {Mn, φn}∞

n=1. Then

φω = (φn)ω ∈ ((Mn)∗)ω = (∏ω Mn)∗. Let p = supp(φω) ∈ ∏ω Mn. .

Theorem (A-Haagerup’12)

. . p(∏ω Mn)p ∼ = (Mn, φn)ω holds. Moreover, (1) σ(φn)ω

t

((xn)ω) = (σφn

t

(xn))ω for (xn)ω ∈ (Mn, φn)ω.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 14 / 24

. . . . . .

Ocneanu vs Groh-Raynaud

M ω ̸= ∏ω M in many ways. But are they unrelated? Consider {Mn, φn}∞

n=1. Then

φω = (φn)ω ∈ ((Mn)∗)ω = (∏ω Mn)∗. Let p = supp(φω) ∈ ∏ω Mn. .

Theorem (A-Haagerup’12)

. . p(∏ω Mn)p ∼ = (Mn, φn)ω holds. Moreover, (1) σ(φn)ω

t

((xn)ω) = (σφn

t

(xn))ω for (xn)ω ∈ (Mn, φn)ω. (2) If M is a type IIIλ (λ ̸= 0) factor, so are M ω, ∏ω M.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 14 / 24

. . . . . .

Ocneanu vs Groh-Raynaud

M ω ̸= ∏ω M in many ways. But are they unrelated? Consider {Mn, φn}∞

n=1. Then

φω = (φn)ω ∈ ((Mn)∗)ω = (∏ω Mn)∗. Let p = supp(φω) ∈ ∏ω Mn. .

Theorem (A-Haagerup’12)

. . p(∏ω Mn)p ∼ = (Mn, φn)ω holds. Moreover, (1) σ(φn)ω

t

((xn)ω) = (σφn

t

(xn))ω for (xn)ω ∈ (Mn, φn)ω. (2) If M is a type IIIλ (λ ̸= 0) factor, so are M ω, ∏ω M. (3) If M is a type III0 factor, then M ω is never a factor.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 14 / 24

. . . . . .

Ocneanu vs Groh-Raynaud

M ω ̸= ∏ω M in many ways. But are they unrelated? Consider {Mn, φn}∞

n=1. Then

φω = (φn)ω ∈ ((Mn)∗)ω = (∏ω Mn)∗. Let p = supp(φω) ∈ ∏ω Mn. .

Theorem (A-Haagerup’12)

. . p(∏ω Mn)p ∼ = (Mn, φn)ω holds. Moreover, (1) σ(φn)ω

t

((xn)ω) = (σφn

t

(xn))ω for (xn)ω ∈ (Mn, φn)ω. (2) If M is a type IIIλ (λ ̸= 0) factor, so are M ω, ∏ω M. (3) If M is a type III0 factor, then M ω is never a factor. (4) If M is a type III1 factor, then any two n.f. states on M ω are unitarily equivalent.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 14 / 24

. . . . . .

Ocneanu vs Groh-Raynaud

M ω ̸= ∏ω M in many ways. But are they unrelated? Consider {Mn, φn}∞

n=1. Then

φω = (φn)ω ∈ ((Mn)∗)ω = (∏ω Mn)∗. Let p = supp(φω) ∈ ∏ω Mn. .

Theorem (A-Haagerup’12)

. . p(∏ω Mn)p ∼ = (Mn, φn)ω holds. Moreover, (1) σ(φn)ω

t

((xn)ω) = (σφn

t

(xn))ω for (xn)ω ∈ (Mn, φn)ω. (2) If M is a type IIIλ (λ ̸= 0) factor, so are M ω, ∏ω M. (3) If M is a type III0 factor, then M ω is never a factor. (4) If M is a type III1 factor, then any two n.f. states on M ω are unitarily equivalent. (5) If M is a type III0 factor, then ∃φn s.t. (M, φn)ω is tracial.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 14 / 24

. . . . . .

Ocneanu vs Groh-Raynaud

M ω ̸= ∏ω M in many ways. But are they unrelated? Consider {Mn, φn}∞

n=1. Then

φω = (φn)ω ∈ ((Mn)∗)ω = (∏ω Mn)∗. Let p = supp(φω) ∈ ∏ω Mn. .

Theorem (A-Haagerup’12)

. . p(∏ω Mn)p ∼ = (Mn, φn)ω holds. Moreover, (1) σ(φn)ω

t

((xn)ω) = (σφn

t

(xn))ω for (xn)ω ∈ (Mn, φn)ω. (2) If M is a type IIIλ (λ ̸= 0) factor, so are M ω, ∏ω M. (3) If M is a type III0 factor, then M ω is never a factor. (4) If M is a type III1 factor, then any two n.f. states on M ω are unitarily equivalent. (5) If M is a type III0 factor, then ∃φn s.t. (M, φn)ω is tracial. (6) Mω ̸= M ′ ∩ M ω in general, but Mω = C ⇒ M ′ ∩ M ω = C if M∗ separable (solution to Ueda’s problem).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 14 / 24

. . . . . .

Approximation Theorem

Connes’s Embedding for general von Neumann algebras? . . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 15 / 24

. . . . . .

Approximation Theorem

Connes’s Embedding for general von Neumann algebras? × Every separable M embeds into B(ℓ2). . . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 15 / 24

. . . . . .

Approximation Theorem

Connes’s Embedding for general von Neumann algebras? × Every separable M embeds into B(ℓ2). We have to add an extra assumption: normal conditional expectation. . . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 15 / 24

. . . . . .

Approximation Theorem

Connes’s Embedding for general von Neumann algebras? × Every separable M embeds into B(ℓ2). We have to add an extra assumption: normal conditional expectation. .

Definition

. . For vNas M, N, we write N

←

֒ →

ε i M if there is an embedding

i : N → M and a normal faithful conditional expectation ε : M → i(N). . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 15 / 24

. . . . . .

Approximation Theorem

Connes’s Embedding for general von Neumann algebras? × Every separable M embeds into B(ℓ2). We have to add an extra assumption: normal conditional expectation. .

Definition

. . For vNas M, N, we write N

←

֒ →

ε i M if there is an embedding

i : N → M and a normal faithful conditional expectation ε : M → i(N). .

Remark

. . If M

←

֒ →

ε i B(ℓ2), then M is atomic.

If N

←

֒ →

ε i M and M is semifinite, then N is also semifinite

(Sakai,Tomiyama).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 15 / 24

. . . . . .

Approximation Theorem

.

Proposition (A-Haagerup-Winsløw ’13)

. . Suppose Mn → N in vN(H). Then for any n.f. state χ ∈ B(H)∗, N

←

֒ →

ε i (Mn, ψn)ω,

where ψn := χ|Mn and moreover φ = (ψn)ω ◦ i, φ := χ|N holds. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 16 / 24

. . . . . .

Approximation Theorem

.

Proposition (A-Haagerup-Winsløw ’13)

. . Suppose Mn → N in vN(H). Then for any n.f. state χ ∈ B(H)∗, N

←

֒ →

ε i (Mn, ψn)ω,

where ψn := χ|Mn and moreover φ = (ψn)ω ◦ i, φ := χ|N holds. .

Sketch.

. . (Construction of i) Let x ∈ N = lim Mn. Then N = lim inf Mn. So ∃(xn)n ∈ ℓ∞(N, Mn) s.t. xn

so∗

→ x. Then (xn)n ∈ Mω(Mn, ψn) and i: N ∋ x → (xn)ω ∈ (Mn, ψn)ω is well-defined.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 16 / 24

. . . . . .

Approximation Theorem

.

Proposition (A-Haagerup-Winsløw ’13)

. . Suppose Mn → N in vN(H). Then for any n.f. state χ ∈ B(H)∗, N

←

֒ →

ε i (Mn, ψn)ω,

where ψn := χ|Mn and moreover φ = (ψn)ω ◦ i, φ := χ|N holds. .

Sketch.

. . (Construction of i) Let x ∈ N = lim Mn. Then N = lim inf Mn. So ∃(xn)n ∈ ℓ∞(N, Mn) s.t. xn

so∗

→ x. Then (xn)n ∈ Mω(Mn, ψn) and i: N ∋ x → (xn)ω ∈ (Mn, ψn)ω is well-defined. (Construction of ε) Given x = (xn)ω ∈ (Mn, ψn)ω, let ˜ x := wo- limω xn. Then as N = lim sup Mn, ˜ x ∈ N and ε: (Mn, ψn)ω ∋ x → i(˜ x) ∈ i(N) is well-defined. By direct calculations, we have N

←

֒ →

ε i (Mn, ψn)ω.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 16 / 24

. . . . . .

.

Corollary (A-Haagerup-Winsløw ’13)

. . Let M, N ∈ vN(H) and assume Mn ∼ = M (∀n) and Mn

n→∞

→ N in vN(H). Then ∃{ψn}∞

n=1 s.t.

N

←

֒ →

ε i (M, ψn)ω

As a (partial) converse: . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 17 / 24

. . . . . .

.

Corollary (A-Haagerup-Winsløw ’13)

. . Let M, N ∈ vN(H) and assume Mn ∼ = M (∀n) and Mn

n→∞

→ N in vN(H). Then ∃{ψn}∞

n=1 s.t.

N

←

֒ →

ε i (M, ψn)ω

As a (partial) converse: .

Proposition (A-Haagerup-Winsløw ’13)

. . Suppose Mn, N ∈ vN(H)st and ∃ψn ∈ Snf(Mn)s.t. N

←

֒ →

ε i (Mn, ψn)ω. Then ∃un ∈ U(H) and ∃n1 < n2 < · · · s.t.

unkMnku∗

nk k→∞

→ N in vN(H).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 17 / 24

. . . . . .

.

Sketch.

. . Put M = (Mn, ψn)ω, ˜ ψ = (ψn)ω, ˜ φ = ˜ ψ ◦ ε. Then by AH’12, ∃φn ∈ Snf(Mn) s.t. M = (Mn, φn)ω and ˜ φ = (φn)ω.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 18 / 24

. . . . . .

.

Sketch.

. . Put M = (Mn, ψn)ω, ˜ ψ = (ψn)ω, ˜ φ = ˜ ψ ◦ ε. Then by AH’12, ∃φn ∈ Snf(Mn) s.t. M = (Mn, φn)ω and ˜ φ = (φn)ω. Let w : L2( M, ˜ ψ) ∋ (xn)ωξ ˜

ψ → (xnξψn)ω ∈ Hω, K := range(w).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 18 / 24

. . . . . .

.

Sketch.

. . Put M = (Mn, ψn)ω, ˜ ψ = (ψn)ω, ˜ φ = ˜ ψ ◦ ε. Then by AH’12, ∃φn ∈ Snf(Mn) s.t. M = (Mn, φn)ω and ˜ φ = (φn)ω. Let w : L2( M, ˜ ψ) ∋ (xn)ωξ ˜

ψ → (xnξψn)ω ∈ Hω, K := range(w).

By Raynaud+AH, ∏ω Mn ↷ Hω standardly, so Jω|K = wJ

Mw∗.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 18 / 24

. . . . . .

.

Sketch.

. . Put M = (Mn, ψn)ω, ˜ ψ = (ψn)ω, ˜ φ = ˜ ψ ◦ ε. Then by AH’12, ∃φn ∈ Snf(Mn) s.t. M = (Mn, φn)ω and ˜ φ = (φn)ω. Let w : L2( M, ˜ ψ) ∋ (xn)ωξ ˜

ψ → (xnξψn)ω ∈ Hω, K := range(w).

By Raynaud+AH, ∏ω Mn ↷ Hω standardly, so Jω|K = wJ

Mw∗.

Since ε( M) = i(N), ˜ φ = φ ◦ ε where φ := ˜ φ|i(N) (Takesaki).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 18 / 24

. . . . . .

.

Sketch.

. . Put M = (Mn, ψn)ω, ˜ ψ = (ψn)ω, ˜ φ = ˜ ψ ◦ ε. Then by AH’12, ∃φn ∈ Snf(Mn) s.t. M = (Mn, φn)ω and ˜ φ = (φn)ω. Let w : L2( M, ˜ ψ) ∋ (xn)ωξ ˜

ψ → (xnξψn)ω ∈ Hω, K := range(w).

By Raynaud+AH, ∏ω Mn ↷ Hω standardly, so Jω|K = wJ

Mw∗.

Since ε( M) = i(N), ˜ φ = φ ◦ ε where φ := ˜ φ|i(N) (Takesaki). So we can identify L2(i(N), φ) ⊂ L2( M, ˜ φ) and ξφ = ξ ˜

φ.

Let L := wL2(i(N), φ). L2( M, ˜ φ)

e ∪

• w

K

eL ∪

• Hω

PK ⊂

• PL
• L2(i(N), φ)

w0

L

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 18 / 24

. . . . . .

.

Sketch.

. . Put M = (Mn, ψn)ω, ˜ ψ = (ψn)ω, ˜ φ = ˜ ψ ◦ ε. Then by AH’12, ∃φn ∈ Snf(Mn) s.t. M = (Mn, φn)ω and ˜ φ = (φn)ω. Let w : L2( M, ˜ ψ) ∋ (xn)ωξ ˜

ψ → (xnξψn)ω ∈ Hω, K := range(w).

By Raynaud+AH, ∏ω Mn ↷ Hω standardly, so Jω|K = wJ

Mw∗.

Since ε( M) = i(N), ˜ φ = φ ◦ ε where φ := ˜ φ|i(N) (Takesaki). So we can identify L2(i(N), φ) ⊂ L2( M, ˜ φ) and ξφ = ξ ˜

φ.

Let L := wL2(i(N), φ). L2( M, ˜ φ)

e ∪

• w

K

eL ∪

• Hω

PK ⊂

• PL
• L2(i(N), φ)

w0

L

By Haagerup-Winsløw’00, ∃vn ∈ U(L, H) s.t. ξ = (vnξ)ω, ∀ξ ∈ L.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 18 / 24

. . . . . .

.

Sketch.

. . Put M = (Mn, ψn)ω, ˜ ψ = (ψn)ω, ˜ φ = ˜ ψ ◦ ε. Then by AH’12, ∃φn ∈ Snf(Mn) s.t. M = (Mn, φn)ω and ˜ φ = (φn)ω. Let w : L2( M, ˜ ψ) ∋ (xn)ωξ ˜

ψ → (xnξψn)ω ∈ Hω, K := range(w).

By Raynaud+AH, ∏ω Mn ↷ Hω standardly, so Jω|K = wJ

Mw∗.

Since ε( M) = i(N), ˜ φ = φ ◦ ε where φ := ˜ φ|i(N) (Takesaki). So we can identify L2(i(N), φ) ⊂ L2( M, ˜ φ) and ξφ = ξ ˜

φ.

Let L := wL2(i(N), φ). L2( M, ˜ φ)

e ∪

• w

K

eL ∪

• Hω

PK ⊂

• PL
• L2(i(N), φ)

w0

L

By Haagerup-Winsløw’00, ∃vn ∈ U(L, H) s.t. ξ = (vnξ)ω, ∀ξ ∈ L. We can then prove that v∗

nMnvn n→ω

→ πL(N) (hard part).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 18 / 24

. . . . . .

.

Sketch.

. . Put M = (Mn, ψn)ω, ˜ ψ = (ψn)ω, ˜ φ = ˜ ψ ◦ ε. Then by AH’12, ∃φn ∈ Snf(Mn) s.t. M = (Mn, φn)ω and ˜ φ = (φn)ω. Let w : L2( M, ˜ ψ) ∋ (xn)ωξ ˜

ψ → (xnξψn)ω ∈ Hω, K := range(w).

By Raynaud+AH, ∏ω Mn ↷ Hω standardly, so Jω|K = wJ

Mw∗.

Since ε( M) = i(N), ˜ φ = φ ◦ ε where φ := ˜ φ|i(N) (Takesaki). So we can identify L2(i(N), φ) ⊂ L2( M, ˜ φ) and ξφ = ξ ˜

φ.

Let L := wL2(i(N), φ). L2( M, ˜ φ)

e ∪

• w

K

eL ∪

• Hω

PK ⊂

• PL
• L2(i(N), φ)

w0

L

By Haagerup-Winsløw’00, ∃vn ∈ U(L, H) s.t. ξ = (vnξ)ω, ∀ξ ∈ L. We can then prove that v∗

nMnvn n→ω

→ πL(N) (hard part). Passing to a subsequence and by the uniqueness of standard form, we are done.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 18 / 24

. . . . . .

With additional efforts, we get the main result: .

Theorem (A-Haagerup-Winsløw ’13)

. . For M ∈ vN(H), and 0 < λ < 1. TFAE. (1) M ∈ Finj. (2) M has QWEP. (3) M

←

֒ →

ε i Rω ∞. R∞: hyperfinite III1 factor.

(4) M

←

֒ →

ε i Rω λ. Rλ: hyperfinite IIIλ factor.

(5) M

←

֒ →

ε i (Mkn(C), φn)ω for some {kn}∞ n=1 and

φn ∈ Snf(Mkn(C)). (6) ∀ε > 0, ∀n ∈ N, ∀ξ1, · · · , ξn ∈ P♮

M, ∃k ∈ N and

∃a1, · · · , an ∈ Mk(C)+ s.t. |⟨ξi, ξj⟩ − trk(aiaj)| < ε (1 ≤ i, j ≤ n). Here, P♮

M is the natural cone in the standard form of M.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 19 / 24

. . . . . .

Very useful trick: .

Theorem (⊗-trick, Haagerup-Winsløw ’00)

. . Let K ∼ = ℓ2, and v0 ∈ U(H ⊗ K, H). Then ∃un ∈ U(H ⊗ K) s.t. for any N ∈ vN(H) and M ∈ vN(K), one has v0u∗

n(N⊗M)u0v∗ n→∞

→ N in vN(H).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 20 / 24

. . . . . .

Very useful trick: .

Theorem (⊗-trick, Haagerup-Winsløw ’00)

. . Let K ∼ = ℓ2, and v0 ∈ U(H ⊗ K, H). Then ∃un ∈ U(H ⊗ K) s.t. for any N ∈ vN(H) and M ∈ vN(K), one has v0u∗

n(N⊗M)u0v∗ n→∞

→ N in vN(H). ⇝ Type III factors are dense, McDuff factors are dense in factors F, etc...

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 20 / 24

. . . . . .

.

Sketch: (1) M ∈ Finj ⇔(3)M

←

֒ →

ε i Rω ∞ ⇒(2) M: QWEP⇒(1).

. . (1)⇒(3) By HW ’00, FIII1 ∩ Finj is dense in Finj, so ∃Mn ∼ = R∞ s.t. Mn → M. Then ∃ψn ∈ Snf(R∞) s.t. N

←

֒ →

ε i (R∞, ψn)ω, but

(R∞, ψn)ω ∼ = Rω

∞ by AH’12.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 21 / 24

. . . . . .

.

Sketch: (1) M ∈ Finj ⇔(3)M

←

֒ →

ε i Rω ∞ ⇒(2) M: QWEP⇒(1).

. . (1)⇒(3) By HW ’00, FIII1 ∩ Finj is dense in Finj, so ∃Mn ∼ = R∞ s.t. Mn → M. Then ∃ψn ∈ Snf(R∞) s.t. N

←

֒ →

ε i (R∞, ψn)ω, but

(R∞, ψn)ω ∼ = Rω

∞ by AH’12.

(3)⇒(1) Assume M

←

֒ →

ε i Rω ∞. Find K1, K2 sep s.t.

• M = M⊗B(K1)⊗C1K2 standard on H ⊗ K, K := K1 ⊗ K2.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 21 / 24

. . . . . .

.

Sketch: (1) M ∈ Finj ⇔(3)M

←

֒ →

ε i Rω ∞ ⇒(2) M: QWEP⇒(1).

. . (1)⇒(3) By HW ’00, FIII1 ∩ Finj is dense in Finj, so ∃Mn ∼ = R∞ s.t. Mn → M. Then ∃ψn ∈ Snf(R∞) s.t. N

←

֒ →

ε i (R∞, ψn)ω, but

(R∞, ψn)ω ∼ = Rω

∞ by AH’12.

(3)⇒(1) Assume M

←

֒ →

ε i Rω ∞. Find K1, K2 sep s.t.

• M = M⊗B(K1)⊗C1K2 standard on H ⊗ K, K := K1 ⊗ K2.

⇝ M

←

֒ →

ε′ i′ Qω, Q := R∞⊗B(K1)⊗C1K2. Since Q,

M: standard, ∃wn s.t. w∗

nQwn →

M, so M ∈ Finj.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 21 / 24

. . . . . .

.

Sketch: (1) M ∈ Finj ⇔(3)M

←

֒ →

ε i Rω ∞ ⇒(2) M: QWEP⇒(1).

. . (1)⇒(3) By HW ’00, FIII1 ∩ Finj is dense in Finj, so ∃Mn ∼ = R∞ s.t. Mn → M. Then ∃ψn ∈ Snf(R∞) s.t. N

←

֒ →

ε i (R∞, ψn)ω, but

(R∞, ψn)ω ∼ = Rω

∞ by AH’12.

(3)⇒(1) Assume M

←

֒ →

ε i Rω ∞. Find K1, K2 sep s.t.

• M = M⊗B(K1)⊗C1K2 standard on H ⊗ K, K := K1 ⊗ K2.

⇝ M

←

֒ →

ε′ i′ Qω, Q := R∞⊗B(K1)⊗C1K2. Since Q,

M: standard, ∃wn s.t. w∗

nQwn →

M, so M ∈ Finj.By ⊗-trick, ∀v0 ∈ U(H ⊗ K, H), ∃un ∈ U(H ⊗ K) s.t. v0u∗

n

Munv∗

0 → M in

vN(H). So M ∈ Finj.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 21 / 24

. . . . . .

.

Sketch: (1) M ∈ Finj ⇔(3)M

←

֒ →

ε i Rω ∞ ⇒(2) M: QWEP⇒(1).

. . (1)⇒(3) By HW ’00, FIII1 ∩ Finj is dense in Finj, so ∃Mn ∼ = R∞ s.t. Mn → M. Then ∃ψn ∈ Snf(R∞) s.t. N

←

֒ →

ε i (R∞, ψn)ω, but

(R∞, ψn)ω ∼ = Rω

∞ by AH’12.

(3)⇒(1) Assume M

←

֒ →

ε i Rω ∞. Find K1, K2 sep s.t.

• M = M⊗B(K1)⊗C1K2 standard on H ⊗ K, K := K1 ⊗ K2.

⇝ M

←

֒ →

ε′ i′ Qω, Q := R∞⊗B(K1)⊗C1K2. Since Q,

M: standard, ∃wn s.t. w∗

nQwn →

M, so M ∈ Finj.By ⊗-trick, ∀v0 ∈ U(H ⊗ K, H), ∃un ∈ U(H ⊗ K) s.t. v0u∗

n

Munv∗

0 → M in

vN(H). So M ∈ Finj. (3)⇒(2) As ∏ω R∞ has QWEP, Rω

∞ ∼

= p(∏ω R∞)p has QWEP too. So by M

←

֒ →

ε i Rω ∞, M has QWEP.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 21 / 24

. . . . . .

.

Sketch: (1) M ∈ Finj ⇔(3)M

←

֒ →

ε i Rω ∞ ⇒(2) M: QWEP⇒(1).

. . (1)⇒(3) By HW ’00, FIII1 ∩ Finj is dense in Finj, so ∃Mn ∼ = R∞ s.t. Mn → M. Then ∃ψn ∈ Snf(R∞) s.t. N

←

֒ →

ε i (R∞, ψn)ω, but

(R∞, ψn)ω ∼ = Rω

∞ by AH’12.

(3)⇒(1) Assume M

←

֒ →

ε i Rω ∞. Find K1, K2 sep s.t.

• M = M⊗B(K1)⊗C1K2 standard on H ⊗ K, K := K1 ⊗ K2.

⇝ M

←

֒ →

ε′ i′ Qω, Q := R∞⊗B(K1)⊗C1K2. Since Q,

M: standard, ∃wn s.t. w∗

nQwn →

M, so M ∈ Finj.By ⊗-trick, ∀v0 ∈ U(H ⊗ K, H), ∃un ∈ U(H ⊗ K) s.t. v0u∗

n

Munv∗

0 → M in

vN(H). So M ∈ Finj. (3)⇒(2) As ∏ω R∞ has QWEP, Rω

∞ ∼

= p(∏ω R∞)p has QWEP too. So by M

←

֒ →

ε i Rω ∞, M has QWEP.

(2)⇒(1) Assume M has QWEP. Use Haagerup-Junge-Xu ’08 reduction method to reduce the problem to tracial case, and then use Kirchberg ’93+our approximation thm! (Continued)

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 21 / 24

. . . . . .

.

Sketch of (2): M QWEP ⇒(1) M ∈ Finj.

. . Assume M: QWEP. Use HJX’08 method: let G = Z[ 1

2] ⊂ R, and

φ ∈ Snf(M). Let φ be the dual state on N := M ⋊σϕ G. Then M

←

֒ →

ε π N.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 22 / 24

. . . . . .

.

Sketch of (2): M QWEP ⇒(1) M ∈ Finj.

. . Assume M: QWEP. Use HJX’08 method: let G = Z[ 1

2] ⊂ R, and

φ ∈ Snf(M). Let φ be the dual state on N := M ⋊σϕ G. Then M

←

֒ →

ε π N. By HJX, ∃N1 ⊂ N2 ⊂ · · · seq of finite vN subalgs of N

with so*-dense union and a n.f. expectation εn : N → Nn s.t.

• φ ◦ εn =

φ, σ

φ t ◦ εn = εn ◦ σ φ t ,

(t ∈ R).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 22 / 24

. . . . . .

.

Sketch of (2): M QWEP ⇒(1) M ∈ Finj.

. . Assume M: QWEP. Use HJX’08 method: let G = Z[ 1

2] ⊂ R, and

φ ∈ Snf(M). Let φ be the dual state on N := M ⋊σϕ G. Then M

←

֒ →

ε π N. By HJX, ∃N1 ⊂ N2 ⊂ · · · seq of finite vN subalgs of N

with so*-dense union and a n.f. expectation εn : N → Nn s.t.

• φ ◦ εn =

φ, σ

φ t ◦ εn = εn ◦ σ φ t ,

(t ∈ R). Let θ = σφ|G, then by Takesaki duality (N ⋊θ G) ∼ = M⊗B(ℓ2) has QWEP, and so does N. ⇝ ∀Nn: QWEP and finite.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 22 / 24

. . . . . .

.

Sketch of (2): M QWEP ⇒(1) M ∈ Finj.

. . Assume M: QWEP. Use HJX’08 method: let G = Z[ 1

2] ⊂ R, and

φ ∈ Snf(M). Let φ be the dual state on N := M ⋊σϕ G. Then M

←

֒ →

ε π N. By HJX, ∃N1 ⊂ N2 ⊂ · · · seq of finite vN subalgs of N

with so*-dense union and a n.f. expectation εn : N → Nn s.t.

• φ ◦ εn =

φ, σ

φ t ◦ εn = εn ◦ σ φ t ,

(t ∈ R). Let θ = σφ|G, then by Takesaki duality (N ⋊θ G) ∼ = M⊗B(ℓ2) has QWEP, and so does N. ⇝ ∀Nn: QWEP and finite. By Kirchberg’93, Nn

←

֒ →

εn in Rω, whence by AHW ’13, Nn ∈ Finj.

Moreover, Nn → N, so N ∈ Finj.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 22 / 24

. . . . . .

.

Sketch of (2): M QWEP ⇒(1) M ∈ Finj.

. . Assume M: QWEP. Use HJX’08 method: let G = Z[ 1

2] ⊂ R, and

φ ∈ Snf(M). Let φ be the dual state on N := M ⋊σϕ G. Then M

←

֒ →

ε π N. By HJX, ∃N1 ⊂ N2 ⊂ · · · seq of finite vN subalgs of N

with so*-dense union and a n.f. expectation εn : N → Nn s.t.

• φ ◦ εn =

φ, σ

φ t ◦ εn = εn ◦ σ φ t ,

(t ∈ R). Let θ = σφ|G, then by Takesaki duality (N ⋊θ G) ∼ = M⊗B(ℓ2) has QWEP, and so does N. ⇝ ∀Nn: QWEP and finite. By Kirchberg’93, Nn

←

֒ →

εn in Rω, whence by AHW ’13, Nn ∈ Finj.

Moreover, Nn → N, so N ∈ Finj.Then by (1)⇒(3), N

←

֒ →

ε j Rω ∞.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 22 / 24

. . . . . .

.

Sketch of (2): M QWEP ⇒(1) M ∈ Finj.

. . Assume M: QWEP. Use HJX’08 method: let G = Z[ 1

2] ⊂ R, and

φ ∈ Snf(M). Let φ be the dual state on N := M ⋊σϕ G. Then M

←

֒ →

ε π N. By HJX, ∃N1 ⊂ N2 ⊂ · · · seq of finite vN subalgs of N

with so*-dense union and a n.f. expectation εn : N → Nn s.t.

• φ ◦ εn =

φ, σ

φ t ◦ εn = εn ◦ σ φ t ,

(t ∈ R). Let θ = σφ|G, then by Takesaki duality (N ⋊θ G) ∼ = M⊗B(ℓ2) has QWEP, and so does N. ⇝ ∀Nn: QWEP and finite. By Kirchberg’93, Nn

←

֒ →

εn in Rω, whence by AHW ’13, Nn ∈ Finj.

Moreover, Nn → N, so N ∈ Finj.Then by (1)⇒(3), N

←

֒ →

ε j Rω ∞.

Then M

←

֒ →

ε0 i0 Rω ∞ so M ∈ Finj by (3)⇒(1).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 22 / 24

. . . . . .

.

Corollary

. . If QWEP conjecture fails, then vN(H)¬QWEP = {M; M ̸= QWEP} is open dense in vN(H). . . . . . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 23 / 24

. . . . . .

.

Corollary

. . If QWEP conjecture fails, then vN(H)¬QWEP = {M; M ̸= QWEP} is open dense in vN(H). .

Proof.

. . By vN(H)QWEP = Finj, vN(H)¬QWEP is open. Assume ∃M non-QWEP and N ∈ vN(H). . . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 23 / 24

. . . . . .

.

Corollary

. . If QWEP conjecture fails, then vN(H)¬QWEP = {M; M ̸= QWEP} is open dense in vN(H). .

Proof.

. . By vN(H)QWEP = Finj, vN(H)¬QWEP is open. Assume ∃M non-QWEP and N ∈ vN(H).By ⊗-Trick, ∃un, ∃v0 s.t. v0u∗

n(N⊗M)unv∗ 0 → N in vN(H)

. . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 23 / 24

. . . . . .

.

Corollary

. . If QWEP conjecture fails, then vN(H)¬QWEP = {M; M ̸= QWEP} is open dense in vN(H). .

Proof.

. . By vN(H)QWEP = Finj, vN(H)¬QWEP is open. Assume ∃M non-QWEP and N ∈ vN(H).By ⊗-Trick, ∃un, ∃v0 s.t. v0u∗

n(N⊗M)unv∗ 0 → N in vN(H) and v0u∗ n(N⊗M)unv∗ 0 fails

• QWEP. So vN(H)¬QWEP is dense.

. . . . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 23 / 24

. . . . . .

.

Corollary

. . If QWEP conjecture fails, then vN(H)¬QWEP = {M; M ̸= QWEP} is open dense in vN(H). .

Proof.

. . By vN(H)QWEP = Finj, vN(H)¬QWEP is open. Assume ∃M non-QWEP and N ∈ vN(H).By ⊗-Trick, ∃un, ∃v0 s.t. v0u∗

n(N⊗M)unv∗ 0 → N in vN(H) and v0u∗ n(N⊗M)unv∗ 0 fails

• QWEP. So vN(H)¬QWEP is dense.

Farah-Hart-Sherman proved by model theory argument, that: .

Theorem (Farah-Hart-Sherman ’11)

. . ∃M ∈ FII1 s.t. N ֒ → M ω for every N ∈ FII1. . . .

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 23 / 24

. . . . . .

.

Corollary

. . If QWEP conjecture fails, then vN(H)¬QWEP = {M; M ̸= QWEP} is open dense in vN(H). .

Proof.

. . By vN(H)QWEP = Finj, vN(H)¬QWEP is open. Assume ∃M non-QWEP and N ∈ vN(H).By ⊗-Trick, ∃un, ∃v0 s.t. v0u∗

n(N⊗M)unv∗ 0 → N in vN(H) and v0u∗ n(N⊗M)unv∗ 0 fails

• QWEP. So vN(H)¬QWEP is dense.

Farah-Hart-Sherman proved by model theory argument, that: .

Theorem (Farah-Hart-Sherman ’11)

. . ∃M ∈ FII1 s.t. N ֒ → M ω for every N ∈ FII1. .

Theorem (A-Haagerup-Winsløw’13)

. . ∃M ∈ FIII1 s.t. N

←

֒ →

ε i M ω for every N ∈ vN(H).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 23 / 24

. . . . . .

.

Proof.

. . Choose (by HW ’00) a sequence {Mn}∞

n=1 of III1 factors which is dense

in vN(H) and φn ∈ Snf(Mn), and put (M, φ) := ⊗

N(Mn, φn).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 24 / 24

. . . . . .

.

Proof.

. . Choose (by HW ’00) a sequence {Mn}∞

n=1 of III1 factors which is dense

in vN(H) and φn ∈ Snf(Mn), and put (M, φ) := ⊗

N(Mn, φn).

Then ∀N, ∃{nk}∞

k=1 s.t. Mnk → N, and (Mnk, φnk) ←

֒ →

εk ik (M, φ).

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 24 / 24

. . . . . .

.

Proof.

. . Choose (by HW ’00) a sequence {Mn}∞

n=1 of III1 factors which is dense

in vN(H) and φn ∈ Snf(Mn), and put (M, φ) := ⊗

N(Mn, φn).

Then ∀N, ∃{nk}∞

k=1 s.t. Mnk → N, and (Mnk, φnk) ←

֒ →

εk ik (M, φ).

Then by approximation thm, one has ∃ψnk ∈ Snf(Mnk) s.t. N

←

֒ →

ε0 i0 (Mnk, ψnk)ω (♡)

∼ = (Mnk, φnk)ω ← ֒ →

(εk)ω (ik)ω (M, φ)ω ∼

= M ω. Here, (♡) is by Connes-Størmer.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 24 / 24

. . . . . .

.

Proof.

. . Choose (by HW ’00) a sequence {Mn}∞

n=1 of III1 factors which is dense

in vN(H) and φn ∈ Snf(Mn), and put (M, φ) := ⊗

N(Mn, φn).

Then ∀N, ∃{nk}∞

k=1 s.t. Mnk → N, and (Mnk, φnk) ←

֒ →

εk ik (M, φ).

Then by approximation thm, one has ∃ψnk ∈ Snf(Mnk) s.t. N

←

֒ →

ε0 i0 (Mnk, ψnk)ω (♡)

∼ = (Mnk, φnk)ω ← ֒ →

(εk)ω (ik)ω (M, φ)ω ∼

= M ω. Here, (♡) is by Connes-Størmer.

• A. Nou proved that q-deformed Araki-Woods algebras have QWEP,

whence they embed into Rω

∞ with a normal faithful conditional

expectation onto its range.

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 24 / 24

. . . . . .

.

Proof.

. . Choose (by HW ’00) a sequence {Mn}∞

n=1 of III1 factors which is dense

in vN(H) and φn ∈ Snf(Mn), and put (M, φ) := ⊗

N(Mn, φn).

Then ∀N, ∃{nk}∞

k=1 s.t. Mnk → N, and (Mnk, φnk) ←

֒ →

εk ik (M, φ).

Then by approximation thm, one has ∃ψnk ∈ Snf(Mnk) s.t. N

←

֒ →

ε0 i0 (Mnk, ψnk)ω (♡)

∼ = (Mnk, φnk)ω ← ֒ →

(εk)ω (ik)ω (M, φ)ω ∼

= M ω. Here, (♡) is by Connes-Størmer.

• A. Nou proved that q-deformed Araki-Woods algebras have QWEP,

whence they embed into Rω

∞ with a normal faithful conditional

expectation onto its range.

Thank you for your attention!

Hiroshi ANDO (Erwin Schr¨

• dinger Institute, Vienna)

Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology ENS Lyon, 27.9.2013 24 / 24