Decomposable Schur multipliers and non-commutative Fourier - PowerPoint PPT Presentation

Decomposable Schur multipliers and non-commutative Fourier multipliers Christoph Kriegler (Clermont-Ferrand), joint work with C´ edric Arhancet (Besan¸ con) Jussieu – 20 October 2015

Regular operators on classical L p (Ω) spaces Let 1 ≤ p ≤ ∞ , and (Ω k , µ k ) be two σ -finite measure spaces ( k = 1 , 2) . An operator T : L p (Ω 1 ) → L p (Ω 2 ) is called positive if for f ∈ L p (Ω 1 ) , f ≥ 0 pointwise, we always have Tf ≥ 0 pointwise. An operator T : L p (Ω 1 ) → L p (Ω 2 ) is called regular if T = T 1 − T 2 + i ( T 3 − T 4 ) with T 1 , T 2 , T 3 , T 4 positive operators. THEOREM: Let T : L p (Ω 1 ) → L p (Ω 2 ) be a regular operator, X a Banach space and S : X → X a bounded operator. Then the tensor product T ⊗ S : L p (Ω 1 ) ⊗ X ⊂ L p (Ω 1 ; X ) → L p (Ω 2 ; X ) extends to a bounded operator on the Bochner space L p (Ω 1 ; X ) with � T ⊗ S � ≤ � T � reg � S � . Here, � T � reg = sup n ∈ N � T ⊗ I ℓ ∞ n � L p (Ω 1 ; ℓ ∞ n ) < ∞ . n ) → L p (Ω 2 ; ℓ ∞

Schatten classes and non-commutative L p ( M ) spaces Let I be a non-empty index set and 1 ≤ p < ∞ . Then the Schatten class S p I is defined to be the class of all compact operators T on ℓ 2 I such that tr(( T ∗ T ) p / 2 ) < ∞ . S ∞ = { compact operators on ℓ 2 I } . I Let M ⊂ B ( H ) be a von Neumann algebra, i.e. weak ∗ closed involutive subalgebra of B ( H ) . Assume that M is equipped with a semifinite faithful normal trace τ : M + → [0 , ∞ ] . Then for 1 ≤ p < ∞ , the non-commutative L p space is defined to be: L p ( M ) = L p ( M , τ ) = completion of 1 p < ∞} . { x ∈ M : � x � L p ( M ) = τ (( x ∗ x ) p / 2 ) L ∞ ( M ) := M . � Ω · d µ ) , and S p For example, L p (Ω) = L p ( L ∞ (Ω) , I = L p ( B ( ℓ 2 I ) , tr) for 1 ≤ p < ∞ .

Completely bounded and completely positive mappings S p I and L p ( M ) are Banach spaces, but even more: Let n ∈ N . Define a norm on M n ⊗ L p ( M ) = M n ( L p ( M )) by [Pisier] � [ x ij ] � M n ( L p ( M )) = sup {� α · x · β � L p ( M n ( M )) : � α � S 2 p n , � β � S 2 p n ≤ 1 } . L p ( M ) is called an operator space equipped with the sequence of norms on M n ( L p ( M )) , n ∈ N . A mapping u : L p ( M 1 ) → L p ( M 2 ) is called completely bounded if the family of mappings u n : M n ( L p ( M 1 )) → M n ( L p ( M 2 )) , [ x ij ] �→ [ u ( x ij )] satisfy � u � cb = sup n ∈ N � u n � < ∞ . Further, u is called completely positive , if all the mappings u n are positive, where x ∈ M n ( L p ( M 1 )) is defined to be positive if x = y ∗ y with y ∈ M n ( L 2 p ( M 1 )) .

Completely positive mappings and classical L p (Ω) spaces PROPOSITION: Let 1 ≤ p ≤ ∞ . Let L p (Ω) be a classical L p space and L p ( M ) a non-commutative one. Then a positive mapping u : L p ( M ) → L p (Ω) is completely positive. Idea of proof: Let a ∈ M n , 1 and [ x ij ] ∈ M n ( L p ( M )) positive. Then a ∗ [ x ij ] a ∈ L p ( M ) is positive, hence u ( a ∗ [ x ij ] a ) ∈ L p (Ω) is positive. Thus, n n � � a ∗ [ u ( x ij )( ω )] a = a i x ij a j )( ω ) = u ( a ∗ [ x ij ] a )( ω ) a i u ( x ij )( ω ) a j = u ( i , j =1 i , j =1 ≥ 0 . Then for a.a. ω ∈ Ω , [ u ( x ij )( ω )] is positive in M n . Now use the fact that [ f ij ] ∈ M n ( L p (Ω)) is positive if and only if [ f ij ( ω )] ∈ M n is a positive matrix for almost all ω ∈ Ω . Hence [ u ( x ij )] is positive in M n ( L p (Ω)) . COROLLARY: Any positive mapping u : L p (Ω) → L p ( M ) is completely positive.

Definition of decomposable mappings DEFINTION: Let 1 ≤ p ≤ ∞ and T : L p ( M 1 ) → L p ( M 2 ) be a bounded linear mapping. Then T is called decomposable if T = T 1 − T 2 + i ( T 3 − T 4 ) with completely positive mappings T 1 , T 2 , T 3 , T 4 . The set of decomposable operators Dec( L p ( M 1 ) , L p ( M 2 )) is a Banach space equipped with the norm � T � d = sup inf {� T 1 � + � T 2 � + � T 3 � + � T 4 � : | λ |≤ 1 λ T = T 1 − T 2 + i ( T 3 − T 4 ) } .

Equivalent norm for decomposable mappings PROPOSITION: Let 1 ≤ p ≤ ∞ and let T : L p ( M 1 ) → L p ( M 2 ) be a bounded linear mapping. Then T is decomposable if and only if there exist v 1 , v 2 : L p ( M 1 ) → L p ( M 2 ) such that the mapping � v 1 � T : S p 2 ( L p ( M 1 )) → S p 2 ( L p ( M 2 )) T ◦ v 2 is completely positive, where T ◦ ( x ) = ( T ( x ∗ )) ∗ . We let � T � dec = inf { max {� v 1 � cb , � v 2 � cb }} , where the infimum runs over all possible v 1 , v 2 . Then � T � d and � T � dec are equivalent on Dec( L p ( M 1 ) , L p ( M 2 )) .

Properties of decomposable mappings PROPOSITION: Let M 1 , M 2 be QWEP von Neumann algebras. Let 1 < p < ∞ . Then any decomposable map T : L p ( M 1 ) → L p ( M 2 ) is completely bounded and � T � cb ≤ � T � dec . In particular, any completely positive mapping T : L p ( M 1 ) → L p ( M 2 ) is completely bounded. THEOREM [Pisier]: Let M 1 , M 2 be hyperfinite von Neumann algebras. Then T : L p ( M 1 ) → L p ( M 2 ) is decomposable if and only if for any operator space E , T ⊗ I E : L p ( M 1 ; E ) → L p ( M 2 ; E ) is bounded. In this case, in fact � T ⊗ I E � ≤ C � T � dec < ∞ , and sup n ∈ N � T ⊗ I M n � L p ( M 1 ; M n ) → L p ( M 2 ; M n ) ∼ = � T � d ∼ = � T � dec .

Decomposable vs. completely bounded mappings PROPOSITION [Haagerup p = ∞ , A.-K.]: Let M have a finite trace τ and u 1 , . . . , u n ∈ M be arbitrary unitaries. Let 1 ≤ p ≤ ∞ . Consider the map T : ℓ p n → L p ( M ) defined by T ( e k ) = u k . Then � T � dec = n 1 − 1 p . Consider now F n the free group of n generators g 1 , g 2 , . . . , g n , and VN ( F n ) the group von Neumann algebra , contained in B ( ℓ 2 ( F n )) , generated by the unitary elements λ s ( f ) = f ( s − 1 · ) . THEOREM [Haagerup p = ∞ , A.-K.]: Let 1 ≤ p ≤ ∞ . Let n ≥ 2 be an integer. The map T n : ℓ p n → L p ( VN ( F n )) , e k �→ λ g k satisfies � T n � cb ≤ (2 √ n − 1) 1 − 1 p and � T n � dec = n 1 − 1 p . In particular, � T n � dec / � T n � cb → ∞ as n → ∞ .

Open questions Question 1: Let R be the hyperfinite factor of type II 1 and let U 1 , . . . , U n ∈ R be a sequence of self-adjoint anticommuting operators. Suppose 1 ≤ p ≤ ∞ . Consider the map T : ℓ p n → L p ( R ) defined by T ( e k ) = U k . What are the values of � T � , � T � dec , � T � cb ? Question 2: Let 1 ≤ p ≤ ∞ . Do we have for every map T : ℓ p 2 → L p ( M ) the equalities � T � = � T � cb = � T � dec ? True for p = ∞ [Haagerup]. Question 3: Let 1 ≤ p ≤ ∞ . Suppose that for every map T : ℓ p 3 → L p ( M ) we have � T � = � T � cb = � T � dec . Is M necessarily hyperfinite? Even open for p = ∞ .

Definition of Schur multipliers Let I be some index set, 1 ≤ p ≤ ∞ , and φ : I × I → C be a bounded function. A mapping M φ : S p I → S p I is called S p -Schur multiplier if it is of the form M φ ([ x ij ]) = [ φ ( i , j ) x ij ] .

Complementation of Schur multipliers THEOREM [A.-K.]: Let I be some index set. For a completely bounded mapping S : S p I → S p I let φ S : I × I → C , ( i , j ) �→ tr( S ( e ij ) e ji ) . Then the linear mapping P I : CB ( S p I ) → CB ( S p I ) , S �→ M φ S has the following properties: 1. P I takes its values in the completely bounded S p -Schur multipliers. 2. P I is contractive. 3. P I ( S ) = S as soon as S is already a cb S p -Schur multiplier. 4. P I ( S ) is completely positive as soon as S is completely positive.

Proof of Complementation of Schur multipliers PROOF: Let ∆ : B ( ℓ 2 I ) → B ( ℓ 2 I ) ⊗ B ( ℓ 2 I ) be the normal ∗ -isomorphism which preserves the traces onto the sub von Neumann algebra ∆( B ( ℓ 2 I )) ⊆ B ( ℓ 2 I ) ⊗ B ( ℓ 2 I ) such that ∆( e ij ) = e ij ⊗ e ij , ( i , j ∈ I ) . Let E be the normal conditional expectation of B ( ℓ 2 I ) ⊗ B ( ℓ 2 I ) onto ∆( B ( ℓ 2 I )) that leaves tr ⊗ tr invariant. For any i , j , k , l ∈ I we have E ( e ij ⊗ e kl ) = δ ik δ jl e ij ⊗ e ij . Set now P I ( S ) = ∆ − 1 E ( S ⊗ Id S p I )∆ . If S completely positive, then also P I ( S ) is. Moreover, I ≤ � ∆ − 1 E ( S ⊗ Id S p � P I ( S ) � cb , S p I )∆ � cb I → S p ≤ � S � cb , S p I . I → S p Finally check that P I ( S ) is a Schur multiplier and P I ( S ) = S if S is already a Schur multiplier.

Consequences of the complementation COROLLARY: Let I be an index set, 1 < p < ∞ and φ : I × I → C a bounded function. Then M φ is a decomposable S p -Schur multiplier if and only if M φ is a bounded Schur multiplier B ( ℓ 2 I ) → B ( ℓ 2 I ) . ⇒ ”: Let M φ : S p I → S p Proof: “= I be decomposable. Then M φ = R 1 − R 2 + i ( R 3 − R 4 ) with completely positive R k . Thus, M φ = P I ( M φ ) = P I ( R 1 ) − P I ( R 2 ) + i ( P I ( R 3 ) − P I ( R 4 )) . Now each P I ( R k ) is a completely positive S p -Schur multiplier, which is known to be bounded on B ( ℓ 2 I ) . Thus, also M φ = P I ( M φ ) is bounded on B ( ℓ 2 I ) . =”: M φ bounded on B ( ℓ 2 “ ⇐ I ) = ⇒ completely bounded on ⇒ decomposable on S p B ( ℓ 2 ⇒ decomposable on B ( ℓ 2 I ) = I ) = I .