A continuum of C -norms on B ( H ) B ( H ) and related tensor - PowerPoint PPT Presentation

A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Definition A pair of C ∗ algebras ( A , B ) will be called a nuclear pair if A ⊗ min B = A ⊗ max B . Recall ( A , B ) nuclear ∀ B ⇔ A nuclear We will concentrate on two fundamental examples B = B ( ℓ 2 ) C = C ∗ ( I F ∞ ) Note that these are both universal but in two different ways injectively for B = B ( ℓ 2 ) projectively for C = C ∗ ( I F ∞ ) Also both are non-nuclear, also B ≃ ¯ B and C ≃ ¯ C A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Kirchberg, Invent, 1993 proved the following fondamental Theorem ( B , C ) is a nuclear pair. Moreover ( A , C ) is a nuclear pair ⇔ A WEP ( A , B ) is a nuclear pair ⇔ A LLP Kirchberg asked two important questions 1) Is ( B , B ) a nuclear pair ? 2) Is ( C , C ) a nuclear pair ? still OPEN For Question 1 : answer is no (Junge-P , GAFA 1995) Theorem (Junge-P , GAFA 1995) If M , N are not nuclear, then the pair ( M , N ) is not nuclear. A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Wassermann (JFA 1976) characterized nuclear von Neumann algebras as finite direct sums of C ⊗ M n with C commutative. Equivalently, he showed ∀ non nuclear von Neumann algebra M � B = M n ⊂ M n ≥ 1 So preceding theorem reduces to the case � M = N = M n n ≥ 1 or equivalently to the case M = N = B A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Let N be a non nuclear von Neumann algebra Wassermann’s result shows that any separable operator space E embeds (completely isometrically) into N i.e. we can replace B ( H ) by N (for operator space theory) we write this E ⊂ N A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Note that Theorem (Haagerup) C ∗ algebra A is WEP IFF A ⊗ ¯ A satisfies � n � n 1 x j ⊗ ¯ 1 x j ⊗ ¯ ∀ n ∀ x j ∈ A � x j � min = � x j � max So this holds for A = B . But nevertheless min � = max on B ⊗ ¯ B ( or on B ⊗ B ) A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Theorem (Ozawa-P) Let M , N be any pair of von Neumann algebras. If M ⊗ min N � = M ⊗ max N i.e. if card { C ∗ − norms on M ⊗ N } > 1 then: card { C ∗ − norms on M ⊗ N } ≥ 2 ℵ 0 A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Theorem (Ozawa-P) Let M , N be any pair of von Neumann algebras. If M ⊗ min N � = M ⊗ max N i.e. if card { C ∗ − norms on M ⊗ N } > 1 then: card { C ∗ − norms on M ⊗ N } ≥ 2 ℵ 0 A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

We say a C ∗ -norm � · � α on B ( ℓ 2 ) ⊗ B ( ℓ 2 ) is admissible if it is invariant under the flip and tensorizes unital completely positive maps Complement: When M = N = B we can find a continuum of admissible C ∗ -norms α on B ⊗ B This produces a continuum of injective tensor product functors in the sense of Kirchberg by inducing each α on A ⊗ B with A ⊂ B B ⊂ B A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Theorem (Ozawa-P) Let � B = M n n ≥ 1 (so that � { x ∈ B | � x � ≤ 1 } = { x ∈ M n | � x � M n ≤ 1 } ) n ≥ 1 and let M ⊂ B ( ℓ 2 ) be a non-nuclear von Neumann algebra. Then the set of { C ∗ − norms on M ⊗ B } has cardinality equal to 2 2 ℵ 0 . A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Let OS n = { E ⊂ B | dim ( E ) = n } We declare E = F if E , F completely isometric Let ( OS n , d cb ) be the metric space formed of all n -dimensional operator spaces equipped with the “distance" d cb ( E , F ) = inf {� u � cb � u − 1 � cb | u : E → F } ∀ E , F , G ∈ OS n d cb ( E , G ) ≤ d cb ( E , F ) d cb ( F , G ) A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Key ingredient Theorem (Junge-P , 1995) For any n > 2 ( OS n , d cb ) is not separable more precisely there is δ > 0 and a family { E i | i ∈ I } ⊂ OS n with card ( I ) = 2 ℵ 0 such that ∀ i � = j ∈ I d cb ( E i , E j ) > 1 + δ We will use a variant: Theorem Assume given for any E ∈ OS n a separable subset C E ⊂ OS n and let d ( E , F ) = max { d cb ( E , C F ) , d cb ( F , C E ) } Then the same holds (by extracting a suitable subfamily and changing δ > 0 ) for the metric d. A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Let A be a C ∗ algebra. Let OS n ( A ) = { E ∈ OS n | E ⊂ A } Various Remarks: • If A is separable, OS n ( A ) is separable F , if A = C ∗ ( I • for any (possibly uncountable) free group I F ) then OS n ( A ) is separable because E ⊂ C ∗ ( I F ) ⇒ E ⊂ C ∗ ( I F ∞ ) and C ∗ ( I F ∞ ) is separable Notation: d SA ( E ) = inf { d cb ( E , F ) | F ⊂ A } A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

C ∗ -norm A , B C ∗ -algebras Let I ⊂ A be an ideal in A C ∗ -norm on ( A / I ) ⊗ B ∀ t ∈ ( A / I ) ⊗ B α ( t ) = � t � ( A ⊗ min B ) / ( I ⊗ min B ) A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Operator space dual For any E ∈ OS n , say E ⊂ B there is an embedding E ∗ ⊂ B such that M n ( E ∗ ) = CB ( E , M n ) isometrically ∀ n for any operator space F CB ( E , F ) = F ⊗ min E ∗ ⊂ B ⊗ min B isometrically A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Lemma (Key Lemma) Let E ∈ OS n Assume M = A / I Consider E ⊂ M and E ∗ ⊂ N Let t E ∈ E ⊗ E ∗ ⊂ ( A / I ) ⊗ N be associated to Id E Then d SA ( E ) ≤ � t E � ( A ⊗ min N ) / ( I ⊗ min N ) Special case from [JP 1995] A = C t E ∈ B ⊗ B d SA ( E ) = � t E � B⊗ max B = � t E � ( A ⊗ min B ) / ( I ⊗ min B ) where we use B ≃ A / I with A = C ∗ ( I F ) for some large enough free group I F A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Proof that ∃ 2 norms on M ⊗ N Pick E ∈ OS n such that d S C ( E ) > 1 F ) so that A / I = M and E ∗ ⊂ N then set A = C ∗ ( I so that t E ∈ E ⊗ E ∗ ⊂ ( A / I ) ⊗ N α ( t ) = � t � ( A ⊗ min N ) / ( I ⊗ min N ) Then α ( t E ) ≥ d S C ( E ) > 1 = � t E � min so α � = � � min and a fortiori � t E � max > 1 � � max � = � � min A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Proof that ∃ 3 norms on M ⊗ N Assume E ⊂ M and E ∗ ⊂ N again F ) → M be onto, let j E : C ∗ < E > → M be inclusion Let q : C ∗ ( I Let now A E = C ∗ < E > ∗ C ∗ ( I F ) we have a surjection Q E = j E ∗ q : A E → M so that A E / I = M Again we set α E ( t ) = � t � ( A E ⊗ min N ) / ( I⊗ min N ) Note OS n ( A ) is separable (because as previously observed can replace I F by I F ∞ ) Thus we can find F such that d SA ( F ) > 1 and hence on the one hand α E ( t F ) ≥ d SA ( F ) > 1 = � t F � min so α E � = � � min but on the other hand α E ( t E ) = 1 but we just proved that � t E � max > 1 so α E � = � � max A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University

Proof that ∃ 2 ℵ 0 norms on M ⊗ N For any E we associate the class C E = { F ⊂ C ∗ < E > ∗ C ∗ ( I F ) } We consider a family { E i | i ∈ I } in OS n with card 2 ℵ 0 and δ > 0 such that ∀ i � = j ∈ I d ( E i , E j ) > 1 + δ where d ( E , F ) = max { d cb ( E , C F ) , d cb ( F , C E ) } Then the preceding reasoning shows that ∀ i � = j ∈ I we have either α E i ( t E j ) > 1 + δ but α E j ( t E j ) = 1 or α E j ( t E i ) > 1 + δ but α E i ( t E i ) = 1 and hence α E i � = α E j A continuum of C ∗ -norms on B ( H ) ⊗ B ( H ) and related tensor products Gilles Pisier Texas A&M University