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

. Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ echal Topology . Hiroshi ANDO Erwin Schr¨ odinger Institute, Vienna ENS Lyon, 27.9.2013 Joint work with Uffe Haagerup and Carl Winsløw (University of Copenhagen) . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 1 / 24

Outline of Talk . . Kirchberg’s QWEP Conjecture 1 . . Effros-Mar´ echal Topology 2 . . Ultraproduct of von Neumann algebras 3 . . Characterizations of QWEP von Neumann Algebras 4 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¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 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. II 1 factor embeds into R ω ? . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 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. II 1 factor embeds into R ω ? In this talk, we discuss how QWEP property is connected to ultraproducts of von Neumann algebras using topological method. . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 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 = id A . (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¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 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 = id A . (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 II 1 factor M admits an embedding into R ω , where R is the hyperfinite II 1 factor. . . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 4 / 24

. . . . Theorem (Kirchberg ’93) . A separable II 1 factor M embeds into R ω if and only if M has QWEP. . . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 5 / 24

. . . . Theorem (Kirchberg ’93) . A separable II 1 factor M embeds into R ω 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 ( ℓ 2 ) = C ∗ ( F ∞ ) ⊗ max B ( ℓ 2 ) . . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 5 / 24

. Theorem (Kirchberg ’93) . A separable II 1 factor M embeds into R ω 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 ( ℓ 2 ) = C ∗ ( F ∞ ) ⊗ max B ( ℓ 2 ) . . Theorem (Junge-Pisier ’95) . B ( ℓ 2 ) ⊗ min B ( ℓ 2 ) ̸ = B ( ℓ 2 ) ⊗ max B ( ℓ 2 ) . . . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 5 / 24

. . . Effros-Mar´ echal Topology Fix H ∼ = ℓ 2 . vN ( H ) =set of all vNas on H . . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 6 / 24

. . . Effros-Mar´ echal Topology Fix H ∼ = ℓ 2 . 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¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 6 / 24

Effros-Mar´ echal Topology Fix H ∼ = ℓ 2 . 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¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 6 / 24

. . . . Definition (Haagerup-Winsløw ’98) . For { M n } ∞ n =1 ⊂ vN ( H ) , define lim sup n →∞ M n and lim inf n →∞ M n by so ∗ → x, ∃ ( x n ) n ∈ ℓ ∞ ( N , M n ) } . n →∞ M n = { x ∈ B ( H ); x n (1) lim inf . . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 7 / 24

. . . . Definition (Haagerup-Winsløw ’98) . For { M n } ∞ n =1 ⊂ vN ( H ) , define lim sup n →∞ M n and lim inf n →∞ M n by so ∗ → x, ∃ ( x n ) n ∈ ℓ ∞ ( N , M n ) } . n →∞ M n = { x ∈ B ( H ); x n (1) lim inf (2) lim sup n →∞ M n =vNa generated by { x ∈ B ( H ); x is a weak-limit point of ∃ ( x n ) n ∈ ℓ ∞ ( N , M n ) } . . . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 7 / 24

. Definition (Haagerup-Winsløw ’98) . For { M n } ∞ n =1 ⊂ vN ( H ) , define lim sup n →∞ M n and lim inf n →∞ M n by so ∗ → x, ∃ ( x n ) n ∈ ℓ ∞ ( N , M n ) } . n →∞ M n = { x ∈ B ( H ); x n (1) lim inf (2) lim sup n →∞ M n =vNa generated by { x ∈ B ( H ); x is a weak-limit point of ∃ ( x n ) n ∈ ℓ ∞ ( N , M n ) } . . . Theorem (Haagerup-Winsløw ’98) . TFAE. (1) M n → M in vN ( H ) . (2) lim inf n →∞ M n = M = lim sup n →∞ M n . ( ) ′ n →∞ M ′ Moreover, lim sup n →∞ M n = lim inf n holds. . . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 7 / 24

. . . Important subsets of vN ( H ) : F : factors, F inj injective factors, vN ( H ) st standardly acting vNas, F II 1 type II 1 factors, etc. . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 8 / 24

Important subsets of vN ( H ) : F : factors, F inj injective factors, vN ( H ) st standardly acting vNas, F II 1 type II 1 factors, etc. . Theorem (Haagerup-Winsløw ’00) . Subset of vN ( H ) Dense in vN ( H ) ? G δ ? F Yes Yes ∪ n ≤ n 0 F I n , n 0 ∈ N No Yes (closed) F I fin * No (but F σ ) F I ∞ * No (but F σ ) F II 1 Yes No F II ∞ Yes No F III 0 Yes No F III λ , λ ∈ (0 , 1) Yes No F III 1 Yes Yes F inj * Yes F st Yes Yes . . . . . . . Hiroshi ANDO (Erwin Schr¨ odinger Institute, Vienna) Ultraproducts, QWEP von Neumann Algebras, and Effros-Mar´ ENS Lyon, 27.9.2013 echal Topology 8 / 24