Some properties of QWEP C ∗ -algebras Eberhard Kirchberg kirchbrg@math.hu-berlin.de U Copenhagen, 2012, Nov 7 and 9. 1 / 30
Sections The original Connes-Conjecture 1 Relatively weakly injective morphisms. 2 Weakly injective C*-algebras 3 QWEP algebras 4 Special cases of the QWEP-conjecture 5 2 / 30
The original Conjecture of A. Connes : A. Connes wrote in proof Theorem 5.1 (= Separable injective II 1 factors N are isomorphic to the hyperfinite II 1 factor R ): ... We now construct an approximate imbedding of N in R . Apparently such an imbedding ought to exist for all II 1 factors because it does for the regular representation of free groups. ... (Perhaps, he deduced such an embedding of F 2 from the residual finiteness of SL (2 , Z )? – Using that the tensor products of a faithful group representation of a countable discrete group G into non-scalar unitaries in R define an embedding of vN ( G ) into R ω ?) 3 / 30
Sections The original Connes-Conjecture 1 Relatively weakly injective morphisms. 2 Weakly injective C*-algebras 3 QWEP algebras 4 Special cases of the QWEP-conjecture 5 4 / 30
Let F := F n denote any free group on n ∈ { 2 , 3 , . . . ; ∞} generators. (Local properties can be also checked with uncountably many generators.) Definition (1) We say that a C*- mono morphism ϕ : A ֒ → B is relatively weakly injective (r.w.i.), or that A is r.w.i. in B , if ϕ satisfies the following (equivalent !) conditions: (rwi,1) ϕ ⊗ id : A ⊗ max C ∗ ( F ) → B ⊗ max C ∗ ( F ) is injective. (rwi,2) ϕ ⊗ id : A ⊗ max C → B ⊗ max C is injective for every C*-algebra C . (rwi,3) There exists a cp contraction V : B → A ∗∗ s.t. V ◦ ϕ = id on A ⊂ A ∗∗ . (rwi,4) There is a normal conditional expectation E from B ∗∗ onto the weak closure of ϕ ( A ) in B ∗∗ , that is extremal among those conditional expectations. 5 / 30
Properties (rwi,1)-(rwi,4) depend from the chosen C*-monomorphism ϕ : Example (2) There are unital r.w.i. and (nuclear) non-r.w.i. embeddings of C ∗ red ( F ) into the (norm-) ultra-power ( O 2 ) ω . 6 / 30
Some (permanence) properties of r.w.i. maps A ֒ → B : (1) For each factorial representation ρ : A → N ⊂ L ( H ) there exists a projection p ∈ M ∞ ( N ) ∼ = L ( ℓ 2 ) ⊗ N , a factorial representation ρ ′ : B → pM ∞ p such that p ≥ 1 N ⊗ e 11 , ρ = (1 N ⊗ e 11 ) ρ ′ ◦ ϕ (under natural identifications) and that the u.c.p. map X ∈ ρ ′ ( B ) ′′ �→ (1 ⊗ p 11 ) X (1 ⊗ p 11 ) is an extreme point in the u.c.p. maps. (2) If J ⊂ A ⊂ B , J ⊳ B and A / J ֒ → B / J is r.w.i. then A ֒ → B is r.w.i. (3) For each C ∗ -algebra B and every separable subspace X ⊆ B there exists a separable sub- C ∗ -algebra A ⊆ B such that X ⊆ A and A ֒ → B is r.w.i. (4) If A n ֒ → B n ( n = 1 , 2 , . . . ) are r.w.i., then c 0 ( A 1 , A 2 , . . . ) ֒ → c 0 ( B 1 , B 2 , . . . ), ℓ ∞ ( A 1 , A 2 , . . . ) ⊂ ℓ ∞ ( B 1 , B 2 , . . . ), � ω A n ⊂ � ω B n and A 1 ⊗ max A 2 ⊗ · · · → B 1 ⊗ max B 2 ⊗ · · · are r.w.i. *-monomorphisms. 7 / 30
(5) If A n − 1 ⊂ A n ⊂ B n ⊂ B n +1 , and A n ֒ → B n is r.w.i. then indlim n A n ֒ → indlim n B n is r.w.i. (6) A ֒ → B ֒ → C r.w.i. implies A ֒ → C r.w.i. → B r.w.i. if and only if A ∗∗ ֒ → B ∗∗ is r.w.i. (7) A ֒ (8) If M ⊂ N is sub-W*-algebra of a W*-algebra N , then M ֒ → N is r.w.i. if and only if there is a (not necessarily normal) conditional expectation E : N → M . (9) If A ⊂ B , ϕ : A ֒ → C is r.w.i. and ϕ extends to a contraction from B into C ∗∗ then A ֒ → B is r.w.i. (10) (N.Ozawa) If A ⊂ B := ℓ ∞ ( M k 1 , M k 2 , . . . ) / c 0 ( M k 1 , M k 2 , . . . ) is a unital simple sub- C ∗ -algebra with unique tracial state τ , such that D τ : A ֒ → N τ is r.w.i., then A ֒ → B is r.w.i. 8 / 30
Question (3) Is the inclusion map C ∗ red ( G ) ֒ → vN ( G ) r.w.i. for every finitely presented (discrete) group G? Question (4) Let A is a simple unital MF-algebra in the sense of B. Blackadar and suppose that A has the Dixmier property. Let τ the unital tracial state on A. When A ֒ → N τ is r.w.i.? Question (5) Let A is a unital separable exact C ∗ -algebra such that A ⊗ min B and A ⊗ max B are finite for each exact unital (separable) MF-algebra B. Is A an MF-algebra? 9 / 30
Sections The original Connes-Conjecture 1 Relatively weakly injective morphisms. 2 Weakly injective C*-algebras 3 QWEP algebras 4 Special cases of the QWEP-conjecture 5 10 / 30
Let A ⊂ L ( H ) a C ∗ -algebra. Definition (6) A is called weakly injective if it has the following equivalent (!) properties: (wi,1) A ֒ → L ( H ) is r.w.i. (wi,2) For every *-monomorphism ϕ : A → B , ϕ is r.w.i. (wi,3) A ⊗ max C ∗ ( F ) = A ⊗ min C ∗ ( F ), in the sense that there is a unique C*-norm on A ⊙ C ∗ ( F ). (wi,4) For every faithful *-representation ρ : A → L ( H ) there exists a c.p. contraction P : L ( H ) → d ( A ) ′′ with P ◦ ρ = ρ . I.e. A has the weak expectation property ( WEP of Ch. Lance). Notice that WEP is a typical C ∗ -algebra property: A ∗∗ is weakly injective if and only if A is nuclear ! 11 / 30
Sections The original Connes-Conjecture 1 Relatively weakly injective morphisms. 2 Weakly injective C*-algebras 3 QWEP algebras 4 Special cases of the QWEP-conjecture 5 12 / 30
Definition (7) A C ∗ -algebra B is has QWEP ( B is a QWEP algebra) if B is a quotient of weakly injective C ∗ -algebra. Some elementary properties of QWEP algebras, that follows step by step from the above given properties of r.w.i. maps and other above listed properties (using the short-exactness of ⊗ max etc.). (q0) Unital A has QWEP, if and only if, for some surjective/any unital C*-morphism ψ : C ∗ ( F ) → A the *-morphism ψ ⊗ max id : C ∗ ( F ) ⊗ max C ∗ ( F ) → A ⊗ max C ∗ ( F ) naturally factorizes over C ∗ ( F ) ⊗ min C ∗ ( F ). (q1) A has QWEP, if and only if, A ∗∗ has QWEP. (q2) A ⊕ B has QWEP, if and only if, A and B have QWEP. (q3) The class of QWEP-algebras is closed under extensions and inductive limites. 13 / 30
(q4) Every QWEP-algebra is the inductive limit of its separable sub- C ∗ -algebras with QWEP. (q5) If A ֒ → B is r.w.i. and B has QWEP then A has QWEP. In particular all hereditary sub- C ∗ -algebras of B have QWEP. (q6) If N is W*-algebra with separable predual N ∗ , then N has QWEP (as a C ∗ -algebra), if and only if, � the central integral decomposition N = N x d µ ( x ) of N into Factors N x – where x is a character of the center of N – has the property that N x has QWEP µ -almost everywhere. (q7) The class of QWEP-algebras is invariant under C*- (respectively W*-) crossed products by actions of amenable groups (or amenable quantum groups). (q8) R ω has QWEP, – because it is a quotient of the weakly injective C ∗ -algebra ℓ ∞ ( R ). 14 / 30
One gets (as a sort of corollary): Theorem (8) Let N a II 1 factor with separable predual N ∗ . TFAE: 1 N has QWEP. 2 N is a sub-C ∗ -algebra of R ω . 3 For each *-morphism γ : C ∗ ( F ∞ ) → N with weakly dense image in N the (pure and positive) functional ρ on C ∗ ( F ∞ ) ⊙ C ∗ ( F ∞ ) given by ρ ( a ⊗ b ) := τ N ( γ ( a ) γ ( b )) (where we use the natural isomorphism C ∗ ( F ) ∼ = C ∗ ( F ) op ) is continuous with respect to � · � min on C ∗ ( F ∞ ) ⊙ C ∗ ( F ∞ ) . 15 / 30
Corollary (9) TFAE: (a) Connes Embedding Problem has positive answer. (b) Every C ∗ -algebra has QWEP (=: QWEP-conjecture) (c) There is only one C ∗ -algebra-norm on C ∗ ( F ) ⊙ C ∗ ( F ) . (d) For each II 1 factor ( N , τ ) , n ∈ N , u 1 , . . . , u n ∈ N unitary, ε > 0 there exist m ∈ N and v 1 , . . . v n ∈ M m with j u k ) − m − 1 Tr ( v ∗ max j , k | τ ( u ∗ j v k )) | < ε . 16 / 30
Sections The original Connes-Conjecture 1 Relatively weakly injective morphisms. 2 Weakly injective C*-algebras 3 QWEP algebras 4 Special cases of the QWEP-conjecture 5 17 / 30
Theorem (N. Ozawa) The C ∗ -algebra L ( ℓ 2 ) ⊗ min L ( ℓ 2 ) is not weakly injective. (It implies that L ( H ) ⊗ min L ( H ) is not weakly injective for each Hilbert space H of infinite dimension.) Question Is L ( ℓ 2 ) ⊗ max L ( ℓ 2 ) a QWEP algebra? Proposition (K.1992) If A is a unital separable C ∗ -algebra with QWEP, then there exists a separable unital block diagonal C ∗ -algebra B ⊂ ℓ ∞ ( U ) (where U := M 2 ⊗ M 3 ⊗ · · · ) such that B op ⊗ max B = B op ⊗ min B and A is a quotient of B. If A op ∼ = A, then one can manage that B op ∼ = B (in addition). 18 / 30
Let U := M 2 ⊗ M 3 ⊗ · · · denote the universal UHF algebra. Remark (13) By the above mentioned claim of Connes holds vN ( F ) ⊂ R ω . (E.g. because SL (2 , Z ) ⊃ F has a faithful group-representation into non-scalar unitaries of ℓ ∞ ( U ) ⊂ R , and – then using that R⊗R = R –, it defines a unitary representation V : SL (2 , Z ) → R ω , such that g ∈ SL (2 , Z ) �→ V ( g ) ∈ R ω satisfies tr ω ( V ( g )) = 0 for g � = e.) A result of Haagerup implies that C ∗ red ( F ) ֒ → vN ( F ) is r.w.i. Together we get that C ∗ red ( F ) has QWEP (and is exact , because F is closed subgroup of the connected Lie group SL 2 ( R ) ). 19 / 30
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