On Matsaev's conjecture for contractions on noncommutative L p -spaces
aa r X i v : . [ m a t h . OA ] N ov On Matsaev’s conjecture for contractions onnoncommutative L p -spaces Cédric Arhancet
Abstract
We exhibit large classes of contractions on noncommutative L p -spaces which satisfy the non-commutative analogue of Matsaev’s conjecture, introduced by Peller, in 1985. In particular, weprove that every Schur multiplier on a Schatten space S p induced by a contractive Schur multiplieron B ( ℓ ) associated with a real matrix satisfy this conjecture. Moreover, we deal with analoguequestions for C -semigroups. Finally, we disprove a conjecture of Peller concerning norms on thespace of complex polynomials arising from Matsaev’s conjecture and Peller’s problem. Indeed, if S denotes the shift on ℓ p and σ the shift on the Schatten space S p , the norms (cid:13)(cid:13) P ( S ) (cid:13)(cid:13) ℓ p −→ ℓ p and (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) can be different for a complex polynomial P . To estimate the norms of functions of operators is an essential task in Operator Theory. In thissubject, V. V. Matsaev stated the following conjecture in 1971, see [Nik1]. For any p ∞ , let ℓ p S −→ ℓ p denote the right shift operator defined by S ( a , a , a , . . . ) = (0 , a , a , a , . . . ) . Conjecture 1.1
Suppose < p < ∞ , p = 2 . Let Ω be a measure space and let L p (Ω) T −→ L p (Ω) be acontraction. For any complex polynomial P , we have (cid:13)(cid:13) P ( T ) (cid:13)(cid:13) L p (Ω) −→ L p (Ω) (cid:13)(cid:13) P ( S ) (cid:13)(cid:13) ℓ p −→ ℓ p . (1.1)It is easy to see that (1.1) holds true for p = 1 and p = ∞ . Moreover, by using the Fouriertransform, it is clear that for p = 2 , (1.1) is a consequence of von Neumann’s inequality. Finally, veryrecently and after the writing of this paper, S. W. Drury [Dr] found a counterexample in the case p = 4 by using computer.For all other values of p , the validity of (1.1) for any contraction is open. It is well-known that (1.1)holds true for any positive contraction, more generally for all operators L p (Ω) T −→ L p (Ω) which admita contractive majorant (cid:0) i.e. there exists a positive contraction ˜ T satisfying | T ( f ) | ˜ T ( | f | ) (cid:1) . Thisfollows from the fact that these operators admit an isometric dilation. We refer the reader to [ALM],[AkS], [CoW], [Kit], [Nik2] and [Pel1] for information and historical background on this question.In 1985, V.V. Peller [Pel2] introduced a noncommutative version of Matsaev’s conjecture for Schat-ten spaces S p = S p ( ℓ ) . Recall that elements of S p can be regarded as infinite matrices indexed by This work is partially supported by ANR 06-BLAN-0015.2000
Mathematics subject classification:
Primary 46L51; Secondary, 46M35, 46L07
Key words and phrases : Matsaev’s conjecture, noncommutative L p -spaces, complex interpolation, Schur multipliers,Fourier multipliers, dilations, semigroups. × N . Thus we define the linear map S p σ −→ S p as the shift ‘from NW to SE’ which maps any matrix(1.2) a a a · · · a a a · · · a a a · · ·· · · · · · · · · · · · to · · · a a · · · a a · · ·· · · · · · · · · · · · . Let S p ( S p ) be the space of all matrices [ a ij ] i,j > with entries a ij in S p , which represent an element ofthe bigger Schatten space S p ( ℓ ⊗ ℓ ) . The algebraic tensor product S p ⊗ S p can be regarded as adense subspace of S p ( S p ) in a natural way. Then the mapping on S p ( S p ) given by (1.2) is an isometry,which is the unique extension of σ ⊗ I S p to the space S p ( S p ) . (See Section 2 below for more detailson these matricial representations.) Peller’s question is as follows. Question 1.2
Suppose < p < ∞ , p = 2 . Let S p T −→ S p be a contraction on the Schatten space S p .Do we have (1.3) (cid:13)(cid:13) P ( T ) (cid:13)(cid:13) S p −→ S p (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) for any complex polynomial P ? Peller observed that (1.3) holds true when T is an isometry or when S p T −→ S p is defined by T ( x ) = axb , where ℓ a −→ ℓ and ℓ b −→ ℓ are contractions.The Schatten spaces S p are basic examples of noncommutative L p -spaces. It is then natural toextend Peller’s problem to this wider context. This leads to the following question. Question 1.3
Suppose < p < ∞ , p = 2 . Let M be a semifinite von Neumann algebra and let L p ( M ) be the associated noncommutative L p -space. Let L p ( M ) T −→ L p ( M ) be a contraction. Do wehave (cid:13)(cid:13) P ( T ) (cid:13)(cid:13) L p ( M ) −→ L p ( M ) (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) (1.4) for any complex polynomial P ? As in the commutative case, it is easy to see that (1.4) holds true when p = 1 , p = 2 or p = ∞ . Themain purpose of this article is to exhibit large classes of contractions on noncommutative L p -spaceswhich satisfy inequality (1.4) for any complex polynomial P . The next theorem gathers some of ourmain results. Theorem 1.4
Suppose < p < ∞ . The following maps satisfy (1.4) for any complex polynomial P .1. A Schur multiplier S p M A −−→ S p induced by a contractive Schur multiplier B (cid:0) ℓ (cid:1) M A −−→ B (cid:0) ℓ (cid:1) associated with a real-valued matrix A .2. A Fourier multiplier L p (cid:0) VN( G ) (cid:1) M t −−→ L p (cid:0) VN( G ) (cid:1) induced by a contractive Fourier multiplier VN( G ) M t −−→ VN( G ) associated with a real valued function G t −→ R , in the case where G is anamenable discrete group G .3. A Fourier multiplier L p (cid:0) VN( F n ) (cid:1) M t −−→ L p (cid:0) VN( F n ) (cid:1) induced by a unital completely positiveFourier multiplier VN( F n ) M t −−→ VN( F n ) associated with a real valued function F n t −→ R , where F n is the free group with n generators ( n ∞ ). Theorem 1.5
Let B (cid:0) ℓ (cid:1) M A −−→ B (cid:0) ℓ (cid:1) be a unital completely positive Schur multiplier with a real-valuedmatrix A . Then there exists a hyperfinite von Neumann algebra M equipped with a semifinite normalfaithful trace, a unital trace preserving ∗ -automorphism M U −→ M , a unital trace preserving one-to-onenormal ∗ -homomorphism B ( ℓ ) J −→ M such that ( M A ) k = E U k J for any integer k > , where M E −→ B (cid:0) ℓ (cid:1) is the canonical faithful normal trace preserving conditionalexpectation associated with J . Theorem 1.6
Let G be a discrete group. Let VN( G ) M t −−→ VN( G ) be a unital completely positiveFourier multiplier associated with a real valued function G t −→ R . Then there exists a von Neumannalgebra M equipped with a faithful finite normal trace, a unital trace preserving ∗ -automorphism M U −→ M , a unital normal trace preserving one-to-one ∗ -homomorphism VN( G ) J −→ M such that, ( M t ) k = E U k J for any integer k > , where M E −→ V N ( G ) is the canonical faithful normal trace preserving conditionalexpectation associated with J . Moreover, if G is amenable or if G = F n ( n ∞ ), the vonNeumann algebra M has the quotient weak expectation property. Various norms on the space of complex polynomials arise from Matsaev’s conjecture and Peller’sproblem, and it is interesting to try to compare them. If p ∞ , note that the space of all diagonalmatrices in S p can be identified with ℓ p . In this regard, the shift operator ℓ p S −→ ℓ p coincides with therestriction of S p σ −→ S p to diagonal matrices. This readily implies that (cid:13)(cid:13) P ( S ) (cid:13)(cid:13) ℓ p −→ ℓ p (cid:13)(cid:13) P ( σ ) (cid:13)(cid:13) S p −→ S p (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) for any complex polynomial P . We will show the following result, which disproves a conjecture dueto Peller [Pel2, Conjecture 2]. Theorem 1.7
Suppose < p < ∞ , p = 2 . Then there exists a complex polynomial P such that (cid:13)(cid:13) P ( S ) (cid:13)(cid:13) ℓ p −→ ℓ p < (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) . To complete this investigation, we will also show that(1.5) (cid:13)(cid:13) P ( σ ) (cid:13)(cid:13) S p −→ S p = (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) = (cid:13)(cid:13) P ( S ) ⊗ Id S p (cid:13)(cid:13) ℓ p ( S p ) −→ ℓ p ( S p ) for any P (the first of these equalities being due to É. Ricard).The paper is organized as follows. In §2, we fix some notations, we give some background on thekey notion of completely bounded maps on noncommutative L p -spaces, we prove the second equalityof (1.5) and we give some preliminary results. In §3, we show that some Fourier multipliers on L p ( R ) and ℓ p Z are bounded but not completely bounded and we prove Theorem 1.7 and the first equalityof (1.5). §4 is devoted to classes of contractions which satisfy noncommutative Matsaev’s inequality(1.4) for any complex polynomial P . In particular we prove Theorems 1.5 and 1.6. In §5, we considera natural analog of Question 1.3 for C -semigroups of contractions. Finally in §6, we exhibit somepolynomials P which always satisfy (1.4) for any contraction T .3 Preliminaries
Let us recall some basic notations. Let T = (cid:8) z ∈ C | | z | = 1 (cid:9) and δ i,j the symbol of Kronecker.If I is an index set and if E is a vector space, we write M I for the space of the I × I matrices withentries in C and M I ( E ) for the space of the I × I matrices with entries in E . If K is another indexset, we have an isomorphism M I ( M K ) = M I × K .Let M be a von Neumann algebra equipped with a semifinite normal faithful trace τ . For p < ∞ the noncommutative L p -space L p ( M ) is defined as follows. If S + is the set of all positive x ∈ M suchthat τ ( x ) < ∞ and S is its linear span, then L p ( M ) is the completion of S with respect to the norm k x k L p ( M ) = τ (cid:0) | x | p (cid:1) p . One sets L ∞ ( M ) = M . We refer to [PiX], and the references therein, for moreinformation on these spaces.Let p < ∞ . If I is an index set and if we equip the space B (cid:0) ℓ I (cid:1) with the operator norm and thecanonical trace Tr , the space L p (cid:0) B ( ℓ I ) (cid:1) identifies to the Schatten-von Neumann class S pI . The space S pI is the space of those compact operators x from ℓ I into ℓ I such that k x k S pI = (cid:0) Tr ( x ∗ x ) p (cid:1) p < ∞ .The space S ∞ I of compact operators from ℓ I into ℓ I is equipped with the operator norm. For I = N ,we simplify the notations, we let S p for S p N . Elements of S pI are regarded as matrices A = [ a ij ] i,j ∈ I of M I . The space S pI ( S pK ) is the space of those compact operators x from ℓ I ⊗ ℓ K into ℓ I ⊗ ℓ K such that k x k S pI ( S pK ) = (cid:0) (Tr ⊗ Tr )( x ∗ x ) p (cid:1) p < ∞ . Elements of S pI ( S pK ) are regarded as matrices of M I ( M K ) .Let M be a von Neumann algebra equipped with a semifinite normal faithful trace τ . If the vonNeumann algebra B (cid:0) ℓ I (cid:1) ⊗ M is equipped with the semifinite normal faithful trace Tr ⊗ τ , the space L p (cid:0) B ( ℓ I ) ⊗ M (cid:1) identifies to a space S pI (cid:0) L p ( M ) (cid:1) of matrices of M I (cid:0) L p ( M ) (cid:1) . Moreover, under thisidentification, the algebraic tensor product S pI ⊗ L p ( M ) is dense in S pI (cid:0) L p ( M ) (cid:1) .Let N be another von Neumann algebra equipped with a semifinite normal faithful trace. If p ∞ , we say that a linear map L p ( M ) T −→ L p ( N ) is completely bounded if Id S p ⊗ T extendsto a bounded operator S p (cid:0) L p ( M ) (cid:1) Id Sp ⊗ T −−−−−→ S p (cid:0) L p ( N ) (cid:1) . In this case, the completely bounded norm k T k cb,L p ( M ) −→ L p ( N ) is defined by(2.1) k T k cb,L p ( M ) −→ L p ( N ) = (cid:13)(cid:13) Id S p ⊗ T (cid:13)(cid:13) S p ( L p ( M )) −→ S p ( L p ( N )) . If Ω is a measure space, the space S p (cid:0) L p (Ω) (cid:1) is isometric to the L p -space L p (Ω , S p ) of S p -valuedfunctions in Bochner’s sense. Thus, if L p (Ω) T −→ L p (Ω) is a linear map, we have(2.2) k T k cb,L p (Ω) −→ L p (Ω) = (cid:13)(cid:13) T ⊗ Id S p (cid:13)(cid:13) L p (Ω ,S p ) −→ L p (Ω ,S p ) . The notion of completely bounded map and the completely bounded norm defined in (2.1) are thesame that these defined in operator space theory, see [EfR], [Pis2] and [Pis4].Now, we let:
Definition 2.1
Let M be a von Neumann algebra equipped with a faithful semifinite normal trace and p ∞ . Let L p ( M ) T −→ L p ( M ) be a contraction. We say that T satisfies the noncommutativeMatsaev’s property if (1.4) holds for any complex polynomial P . We denote by ℓ p S −→ ℓ p the right shift on ℓ p . We use the same notation for the right shift on ℓ p Z . Wedenote by S − the left shift on ℓ p defined by S − ( a , a , a , . . . ) = ( a , a , a , . . . ) . Suppose p ∞ .Let X be a Banach space. For any complex polynomial P , we define k P k p,X by k P k p,X = (cid:13)(cid:13) P ( S ) ⊗ Id X (cid:13)(cid:13) ℓ p ( X ) −→ ℓ p ( X ) .
4e let k P k p = k P k p, C (cid:0) = k P ( S ) k ℓ p −→ ℓ p (cid:1) . If p < ∞ , it is easy to see that, for any complexpolynomial P , we have(2.3) k P k p,X = (cid:13)(cid:13) P ( S ) ⊗ Id X (cid:13)(cid:13) ℓ p Z ( X ) −→ ℓ p Z ( X ) = (cid:13)(cid:13) P ( S − ) ⊗ Id X (cid:13)(cid:13) ℓ p ( X ) −→ ℓ p ( X ) . Moreover, for all p < ∞ , by (2.2), we have k P k p,S p = (cid:13)(cid:13) P ( S ) (cid:13)(cid:13) cb,ℓ p Z −→ ℓ p Z . (2.4)Note that, if p ∞ , we have k P k p,S p = k P k p ∗ ,S p ∗ . Moreover, if p q , we have k P k q,S q k P k p,S p by interpolation. We define the linear map S p Z Θ −→ S p Z as the shift "from NW toSE" which maps any matrix · · · · · · · · · · · · · · ·· · · a , a , a , · · ·· · · a , a , a , · · ·· · · a , a , a , · · ·· · · · · · · · · · · · · · · to · · · · · · · · · · · · · · ·· · · a − , − a − , a − , · · ·· · · a , − a , a , · · ·· · · a , − a , a , · · ·· · · · · · · · · · · · · · · . If p < ∞ , it is not difficult to see that for any complex polynomial P we have(2.5) (cid:13)(cid:13) P (Θ) (cid:13)(cid:13) S p Z −→ S p Z = (cid:13)(cid:13) P ( σ ) (cid:13)(cid:13) S p −→ S p and (cid:13)(cid:13) P (Θ) (cid:13)(cid:13) cb,S p Z −→ S p Z = (cid:13)(cid:13) P ( σ ) (cid:13)(cid:13) cb,S p −→ S p . Moreover, it is easy to see that, for all A ∈ S p Z , we have the equality Θ( A ) = SAS − where we consider A and Θ( A ) as operators on ℓ Z .We will use the following theorem inspired by a well-known technique of Kitover. Theorem 2.2
Suppose p ∞ . Let X be a Banach space and X T −→ X an isometry (notnecessarily onto). For any complex polynomial P , we have the inequality (cid:13)(cid:13) P ( T ) (cid:13)(cid:13) X −→ X k P k p,X . Proof : It suffices to consider the case < p < ∞ . Let < r < . Since T is an isometry we have + ∞ X j =0 (cid:13)(cid:13) r j T j ( x ) (cid:13)(cid:13) pX = + ∞ X j =0 r jp (cid:13)(cid:13) T j ( x ) (cid:13)(cid:13) pX = k x k pX + ∞ X j =0 ( r p ) j ! < + ∞ . We let C r = + ∞ X j =0 r jp ! p . Now we define the operator W r : X −→ ℓ p ( X ) x C r (cid:0) x, rT ( x ) , r T ( x ) , . . . , r j T j ( x ) , . . . (cid:1) which is an isometry. If n is a positive integer and if x ∈ X we have W r (cid:0) ( rT ) n x (cid:1) = 1 C r (cid:0) r n T n x, r n +1 T n +1 x, . . . (cid:1) = ( S − ⊗ Id X ) n (cid:0) W r ( x ) (cid:1) .
5e deduce that for any complex polynomial P we have W r P ( rT ) = P ( S − ⊗ Id X ) W r . Now, if x ∈ X ,we have (cid:13)(cid:13) P ( rT ) x (cid:13)(cid:13) X = (cid:13)(cid:13) W r (cid:0) P ( rT ) x (cid:1)(cid:13)(cid:13) ℓ p ( X ) = (cid:13)(cid:13) P ( S − ⊗ Id X ) W r ( x ) (cid:13)(cid:13) ℓ p ( X ) (cid:13)(cid:13) P ( S − ) ⊗ Id X (cid:13)(cid:13) ℓ p ( X ) −→ ℓ p ( X ) k x k X = k P k p,X k x k X by (2.3) . Consequently, letting r to , we obtain finally that (cid:13)(cid:13) P ( T ) (cid:13)(cid:13) X −→ X k P k p,X . Corollary 2.3
Suppose p ∞ . Let P be a complex polynomial. We have (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) = (cid:13)(cid:13) P (cid:13)(cid:13) p,S p . Proof : With the diagonal embedding of ℓ p in S p , we see that for any complex polynomial P we have (cid:13)(cid:13) P (cid:13)(cid:13) p,S p (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) . Now the map S p ( S p ) σ ⊗ Id Sp −−−−−→ S p ( S p ) is an isometry. Hence, by the above theorem, we deduce thatfor every complex polynomial P we have (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) = (cid:13)(cid:13) P ( σ ⊗ Id S p ) (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) k P k p,S p ( S p ) = k P k p,S p . Let M be a von Neumann algebra. Let us recall that M has QWEP means that M is the quotient ofa C ∗ -algebra having the weak expectation property (WEP) of C. Lance (see [Oza] for more informationon these notions). It is unknown whether every von Neumann algebra has this property. We will needthe following theorem which is a particular case of a result of [Jun]. Theorem 2.4
Let M be a von Neumann algebra with QWEP equipped with a faithful semifinitenormal trace. Suppose < p < ∞ . Let Ω be a measure space. Suppose that L p (Ω) T −→ L p (Ω) is acompletely bounded map. Then T ⊗ Id L p ( M ) extends to a bounded operator and we have (cid:13)(cid:13) T ⊗ Id L p ( M ) (cid:13)(cid:13) L p (Ω ,L p ( M )) −→ L p (Ω ,L p ( M )) k T k cb,L p (Ω) −→ L p (Ω) . In the case where M is a hyperfinite von Neumann algebra, the statement of this theorem is easy toprove (cid:0) use [Pis2, (3.1)] and [Pis2, (3.6)] (cid:1) . With this theorem, we deduce the following proposition. Proposition 2.5
Suppose < p < ∞ . Let M be a von Neumann algebra with QWEP equipped witha faithful semifinite normal trace. For all complex polynomial P we have k P k p,L p ( M ) k P k p,S p . With this proposition, we can prove the following corollary.
Corollary 2.6
Let M be a von Neumann algebra equipped with a faithful semifinite normal trace and < p < ∞ . Let L p ( M ) T −→ L p ( M ) be a contraction. Suppose that there exists a von Neumann algebra M with QWEP equipped with a faithful semifinite normal trace, an isometric embedding L p ( M ) J −→ L p ( N ) , an isometry L p ( N ) U −→ L p ( N ) and a contractive projection L p ( N ) Q −→ L p ( M ) such that, T k = QU k J for any integer k > . Then the contraction T has the noncommutative Matsaev’s property. roof : For any complex polynomial P , we have (cid:13)(cid:13) P ( T ) (cid:13)(cid:13) L p ( M ) −→ L p ( M ) = (cid:13)(cid:13) QP ( U ) J (cid:13)(cid:13) L p ( M ) −→ L p ( M ) (cid:13)(cid:13) P ( U ) (cid:13)(cid:13) L p ( N ) −→ L p ( N ) . By using Theorem 2.2, we obtain the inequality (cid:13)(cid:13) P ( T ) (cid:13)(cid:13) L p ( M ) −→ L p ( M ) k P k p,L p ( N ) . Now, the von Neumann algebra N is QWEP. Then, by Proposition 2.5, we obtain finally that (cid:13)(cid:13) P ( T ) (cid:13)(cid:13) L p ( M ) −→ L p ( M ) k P k p,S p . Theorem 2.4, Proposition 2.5 and Corollary 2.6 hold true more generally for noncommutative L p -spaces of a von Neumann algebra equipped with a distinguished normal faithful state M ϕ −→ C ,constructed by Haagerup. See [PiX] and the references therein for more informations on these spaces.We refer to [ALM], [AkS], [JLM] and [Pel1] for information on dilations on L p -spaces (commutativeand noncommutative). Suppose < p < ∞ . Let G be a locally compact abelian group with dual group b G . An operator L p ( G ) T −→ L p ( G ) is a Fourier multiplier if there exists a function ψ ∈ L ∞ (cid:0) b G (cid:1) such that for any f ∈ L p ( G ) ∩ L ( G ) we have F (cid:0) T ( f ) (cid:1) = ψ F ( f ) where F denotes the Fourier transform. In this case,we let T = M ψ . G. Pisier showed that, if G is a compact group and < p < ∞ , p = 2 , there exists abounded Fourier multiplier L p ( G ) T −→ L p ( G ) which is not completely bounded (see [Pis2, Proposition8.1.3]. We will show this result is also true for the groups R and Z and we will prove Theorem 1.7.If b ∈ L ( G ) , we define the convolution operator C b by C b : L p ( G ) −→ L p ( G ) f b ∗ f. This operator is a completely bounded Fourier multiplier. We observe that, if P = P nk =0 a k z k isa complex polynomial, the operator ℓ p Z P ( S ) −−−→ ℓ p Z is the operator ℓ p Z C ˜ a −−→ ℓ p Z where ˜ a is the sequencedefined by ˜ a k = a k if k n and ˜ a k = 0 otherwise.We will use the following approximation result [Lar, Theorem 5.6.1]. Theorem 3.1
Suppose p < ∞ . Let G be a locally compact abelian group. Let L p ( G ) T −→ L p ( G ) be a bounded Fourier multiplier. Then there exists a net of continuous functions ( b l ) i ∈ L with compactsupport such that (cid:13)(cid:13) C b l (cid:13)(cid:13) L p ( G ) −→ L p ( G ) k T k L p ( G ) −→ L p ( G ) and C b l so −→ l T (convergence for the strong operator topology). Moreover, we need the following vectorial extension of [DeL, Proposition 3.3]. One can prove thistheorem as [CoW, Theorem 3.4]. 7 heorem 3.2
Suppose < p < ∞ . Let ψ be a continuous function on R which defines a completelybounded Fourier multiplier M ψ on L p ( R ) . Then the restriction ψ | Z of the function ψ to Z defines acompletely bounded Fourier multiplier M ψ | Z on L p ( T ) . We will use the next result of Jodeit [Jod, Theorem 3.5]. We introduce the function
Λ : R −→ R definedby Λ( x ) = (cid:26) − | x | if x ∈ [ − , | x | > . Theorem 3.3
Suppose < p < ∞ . Let ϕ be a complex function defined on Z such that M ϕ is abounded Fourier multiplier on L p ( T ) . Then the complex function R ψ −→ C defined on R by (3.1) ψ ( x ) = X k ∈ Z ϕ ( k )Λ( x − k ) , x ∈ R , defines a bounded Fourier multiplier L p ( R ) M ψ −−→ L p ( R ) . Now, we are ready to prove the following theorem.
Theorem 3.4
Suppose < p < ∞ , p = 2 . Then there exists a bounded Fourier multiplier L p ( R ) M ψ −−→ L p ( R ) which is not completely bounded.Proof : By [Pis2, Proposition 8.1.3], there exists a bounded Fourier multiplier L p ( T ) M ϕ −−→ L p ( T ) which is not completely bounded. Now, we define the function ψ on R by (3.1). By Theorem 3.3, thefunction R ψ −→ C defines a bounded Fourier multiplier L p ( R ) M ψ −−→ L p ( R ) . Now, suppose that M ψ iscompletely bounded. Since the function R ψ −→ C is continuous, by Theorem 3.2, we deduce that therestriction ψ | Z defines a completely bounded Fourier multiplier M ψ | Z on L p ( T ) . Moreover, we observethat, for all k ∈ Z , we have ψ ( k ) = ϕ ( k ) . Then we deduce that the Fourier multiplier L p ( T ) M ϕ −−→ L p ( T ) is completely bounded. We obtaina contradiction. Consequently, the bounded Fourier multiplier L p ( R ) M ψ −−→ L p ( R ) is not completelybounded.The proof of the next theorem is inspired by [CoW, page 25]. Theorem 3.5
Suppose < p < ∞ , p = 2 . Then1. There exists a bounded Fourier multiplier ℓ p Z T −→ ℓ p Z which is not completely bounded.2. There exists a complex polynomial P such that k P k p < k P k p,S p .Proof : By Theorem 3.4, there exists a bounded Fourier multiplier L p ( R ) M ψ −−→ L p ( R ) which is notcompletely bounded. We can suppose that M ψ satisfies (cid:13)(cid:13) M ψ (cid:13)(cid:13) L p ( R ) −→ L p ( R ) = 1 . By Theorem 3.1,there exists a net of continuous functions ( b l ) l ∈ L with compact support such that (cid:13)(cid:13) C b l (cid:13)(cid:13) L p ( R ) −→ L p ( R ) and C b l so −→ l M ψ . Let c > . There exists an element y = P nk =1 f k ⊗ x k ∈ L p ( R ) ⊗ S p with k y k L p ( R ,S p ) such that (cid:13)(cid:13) ( M ψ ⊗ Id S p )( y ) (cid:13)(cid:13) L p ( R ,S p ) > c . Then, it is not difficult to see that there exists l ∈ L such that8 (cid:13) ( C b l ⊗ Id S p )( y ) (cid:13)(cid:13) L p ( R ,S p ) > c . We deduce that there exists a continuous function b : R −→ C withcompact support such that k C b k L p ( R ) −→ L p ( R ) and (cid:13)(cid:13) C b (cid:13)(cid:13) cb,L p ( R ) −→ L p ( R ) > c . Thus there exists acontinuous function R b −→ C with compact support such that k C b k L p ( R ) −→ L p ( R ) and k C b k cb,L p ( R ) −→ L p ( R ) > c. Now, we define the sequence (cid:0) a n (cid:1) n > of complex sequences indexed by Z by, if n > and k ∈ Z a n,k = Z Z n b (cid:18) t − s + kn (cid:19) dsdt. Note that each sequence a n has only a finite number of non-zero term. Let n > . We introduce theconditional expectation L p ( R ) E n −−→ L p ( R ) with respect to the σ -algebra generated by the h kn , k +1 n h , k ∈ Z . For every integer n > and all f ∈ L p ( R ) , we have E n f = n X k ∈ Z Z k +1 nkn f ( t ) dt ! (cid:2) kn , k +1 n (cid:2) (see [AbA, page 227]). Now, we define the linear map ℓ p Z J n −−→ E n (cid:0) L p ( R ) (cid:1) by, if u ∈ ℓ p Z J n ( u ) = n p X k ∈ Z u k (cid:2) kn , k +1 n (cid:2) . It is easy to check that the map J n is an isometry of ℓ p Z onto the range E n (cid:0) L p ( R ) (cid:1) of E n . For any u ∈ ℓ p Z , we have E n C b J n ( u ) = n X k ∈ Z Z k +1 nkn (cid:0) C b J n ( u ) (cid:1) ( t ) dt ! (cid:2) kn , k +1 n (cid:2) = n X k ∈ Z Z k +1 nkn Z + ∞−∞ b ( t − s ) (cid:0) J n ( u ) (cid:1) ( s ) dsdt ! (cid:2) kn , k +1 n (cid:2) = n X k ∈ Z Z k +1 nkn Z + ∞−∞ b ( t − s ) n p X j ∈ Z u j (cid:2) jn , j +1 n (cid:2) ( s ) ! dsdt ! (cid:2) kn , k +1 n (cid:2) (3.2) = n p X k ∈ Z X j ∈ Z u j Z k +1 nkn Z j +1 njn b ( t − s ) dsdt ! (cid:2) kn , k +1 n (cid:2) (3.3) = n p X k ∈ Z X j ∈ Z u j Z n Z n b (cid:18) t − s + k − jn (cid:19) dsdt ! (cid:2) kn , k +1 n (cid:2) = n p X k ∈ Z X j ∈ Z u j Z Z b (cid:18) t − s + k − jn (cid:19) dsdt ! (cid:2) kn , k +1 n (cid:2) = n p X k ∈ Z X j ∈ Z u j a n,k − j ! (cid:2) kn , k +1 n (cid:2) = J n C a n ( u ) j ∈ Z of (3.2) is finite). Thuswe have the following commutative diagram L p ( R ) C b / / L p ( R ) E n (cid:15) (cid:15) E n (cid:0) L p ( R ) (cid:1) ?(cid:31) O O E n (cid:0) L p ( R ) (cid:1) ℓ p Z J n ≈ O O C an / / ℓ p Z . J n ≈ O O Then, for any integer n > , since k E n k L p ( R ) −→ L p ( R ) , we have the following estimate (cid:13)(cid:13) C a n (cid:13)(cid:13) ℓ p Z −→ ℓ p Z k C b k L p ( R ) −→ L p ( R ) . Moreover, we have E n ⊗ Id S p so −−−−−→ n → + ∞ Id L p ( R ,S p ) (see [Cha, Theorem 1]). It is easy to see that ( E n C b E n ) ⊗ Id S p so −−−−−→ n → + ∞ C b ⊗ Id S p . By the strong semicontinuity of the norm, we obtain that k C b k cb,L p ( R ) −→ L p ( R ) lim inf n →∞ (cid:13)(cid:13) E n C b E n (cid:13)(cid:13) cb,L p ( R ) −→ L p ( R ) . Then, there exists an integer n > such that (cid:13)(cid:13) C a n (cid:13)(cid:13) ℓ p Z −→ ℓ p Z and (cid:13)(cid:13) C a n (cid:13)(cid:13) cb,ℓ p Z −→ ℓ p Z > c. Thus, we prove the second assertion by shifting the obtained multiplier. Finally, we show the firstassertion by the closed graph theorem, (2.3) for X = C and (2.4).The paper [Arh] is a continuation of these investigations. The author proves that if G is anarbitrary infinite locally compact abelian group, < p < ∞ and p = 2 then there exists a boundedFourier multiplier on L p ( G ) which is not completely bounded.In the light of Corollary 2.3 and Theorem 3.5, it is natural to compare (cid:13)(cid:13) P ( σ ) (cid:13)(cid:13) cb,S p −→ S p and (cid:13)(cid:13) P ( σ ) (cid:13)(cid:13) S p −→ S p . We finish the section by proving that these quantities are identical. It is a result dueto É. Ricard. In order to prove it, we need the following notion of Schur multiplier. We equip T withits normalized Haar measure. We denote by S p (cid:0) L ( T ) (cid:1) the Schatten-von Neumann class associatedwith B (cid:0) L ( T ) (cid:1) . If f ∈ L ( T × T ) , we denote the associated Hilbert-Schmidt operator by K f : L ( T ) −→ L ( T ) u R T u ( z ) f ( z, · ) dz. A Schur multiplier on S p (cid:0) L ( T ) (cid:1) is a linear map S p (cid:0) L ( T ) (cid:1) T −→ S p (cid:0) L ( T ) (cid:1) such that there exists ameasurable function T × T ϕ −→ C which satisfies, for any finite rank operator of the form L ( T ) K f −−→ L ( T ) , the equality T ( K f ) = K ϕf . We denote T by M ϕ and we say that the function ϕ is the symbolof the Schur multiplier S p (cid:0) L ( T ) (cid:1) M ϕ −−→ S p (cid:0) L ( T ) (cid:1) (see [BiS] and [LaS] for more details).We denote by L ( T ) F −→ ℓ Z the Fourier transform. We define the isometry Ψ by Ψ : S p (cid:0) L ( T ) (cid:1) −→ S p Z T T F − . Now, we can show the following proposition. 10 roposition 3.6
Suppose p ∞ . For any complex polynomial P , we have (cid:13)(cid:13) P ( σ ) (cid:13)(cid:13) S p −→ S p = (cid:13)(cid:13) P ( σ ) (cid:13)(cid:13) cb,S p −→ S p (cid:16) = (cid:13)(cid:13) P ( σ ) ⊗ Id S p (cid:13)(cid:13) S p ( S p ) −→ S p ( S p ) (cid:17) . Proof : It suffices to consider the case < p < ∞ . For any n ∈ Z and any finite rank operator of theform K f , we have (cid:16) ΘΨ (cid:0) K f (cid:1)(cid:17) ( e n ) = S F K f F − S − ( e n )= S F K f ( z n − )= S F Z T z n − f ( z, · ) dz ! = S X k ∈ Z Z T z n z ′ k f ( z, z ′ ) dzdz ′ ! e k ! = X k ∈ Z Z T z n − z ′ k f ( z, z ′ ) dzdz ′ ! e k +1 . Now we define the function T × T ϕ −→ C by ϕ ( z, z ′ ) = z − z ′ where z, z ′ ∈ T . Then, for any n ∈ Z andany finite rank operator of the form K f , we have (cid:16) Ψ M ϕ (cid:0) K f (cid:1)(cid:17) ( e n ) = F K ϕf F − ( e n )= F Z T z n ϕ ( z, · ) f ( z, · ) dz ! = X k ∈ Z Z T z n z ′ k ϕ ( z, z ′ ) f ( z, z ′ ) dzdz ′ ! e k = X k ∈ Z Z T z n − z ′ k − f ( z, z ′ ) dzdz ′ ! e k = X k ∈ Z Z T z n − z ′ k f ( z, z ′ ) dzdz ′ ! e k +1 . Then, for any complex polynomial P , we have the following commutative diagram S p Z P (Θ) / / S p Z S p (cid:0) L ( T ) (cid:1) Ψ O O P ( M ϕ ) / / S p (cid:0) L ( T ) (cid:1) . Ψ O O Furthermore, for any complex polynomial P , we have P ( M ϕ ) = M P ( ϕ ) . Moreover, the Schur multiplier S p (cid:0) L ( T ) (cid:1) M P ( ϕ ) −−−−→ S p (cid:0) L ( T ) (cid:1) has a continuous symbol whose the support has no isolated point. By[LaS, Theorem 1.19], we deduce that the norm and the completely bounded norm of P ( M ϕ ) coincide.Since Ψ is a complete isometry, we obtain the result by (2.5).11 Positive results
Let M and N be von Neumann algebras equipped with faithful semifinite normal traces τ M and τ N .Let M T −→ N a positive linear map. We say that T is trace preserving if for all x ∈ L ( M ) ∩ M + wehave τ N (cid:0) T ( x ) (cid:1) = τ M ( x ) . We will use the following straightforward extension of [JuX, Lemma 1.1]. Lemma 4.1
Let M and N be von Neumann algebras equipped with faithful semifinite normal traces.Let M T −→ N be a trace preserving unital normal positive map. Suppose p < ∞ . Then T induces acontraction L p ( M ) T −→ L p ( N ) . Moreover, if M T −→ N is an one-to-one normal unital ∗ -homomorphism, T induces an isometry L p ( M ) T −→ L p ( N ) . Let M be a von Neumann algebra equipped with faithful semifinite normal trace τ and N a vonNeumann subalgebra such that the restriction of τ is still semifinite. Then, it is well-known that theextension L p ( M ) E −→ L p ( N ) of the canonical faithful normal trace preserving conditional M E −→ N is acontractive projection.Consider the situation where M T −→ M is a linear map such there exists a von Neumann algebra N equipped with a faithful semifinite normal trace, a unital trace preserving ∗ -automorphism N U −→ N ,a unital normal trace preserving one-to-one ∗ -homomorphism M J −→ N such that, T k = E U k J (4.1)for any integer k > , where N E −→ M is the canonical faithful normal trace preserving conditionalexpectation associated with J . Then, for all p < ∞ , the maps N U −→ N and M J −→ N extendto isometries L p ( N ) U −→ L p ( N ) and L p ( M ) J −→ L p ( N ) and the map N E −→ M extends to a contractiveprojection L p ( N ) E −→ L p ( M ) such that (4.1) is also true for the induced map L p ( M ) T −→ L p ( M ) .In order to prove Theorems 1.5 and 1.6, we need to use fermion algebras. Since we will study mapsbetween q -deformed algebras, we recall directly several facts about these more general algebras in thecontext of [BKS]. We denote by S n the symmetric group. If σ is a permutation of S n we denote by | σ | the number card (cid:8) ( i, j ) | i, j n, σ ( i ) > σ ( j ) (cid:9) of inversions of σ . Let H be a real Hilbert spacewith complexification H C . If − q < the q -Fock space over H is F q ( H ) = C Ω ⊕ M n > H ⊗ n C where Ω is a unit vector, called the vacuum and where the scalar product on H ⊗ n C is given by h h ⊗ · · · ⊗ h n , k ⊗ · · · ⊗ k n i q = X σ ∈ S n q | σ | h h , k σ (1) i H C · · · h h n , k σ ( n ) i H C . If q = − , we must first divide out by the null space, and we obtain the usual antisymmetric Fockspace. The creation operator l ( e ) for e ∈ H is given by l ( e ) : F q ( H ) −→ F q ( H ) h ⊗ · · · ⊗ h n e ⊗ h ⊗ · · · ⊗ h n . They satisfy the q -relation l ( f ) ∗ l ( e ) − ql ( e ) l ( f ) ∗ = h f, e i H Id F q ( H ) .
12e denote by F q ( H ) ω ( e ) −−−→ F q ( H ) the selfadjoint operator l ( e ) + l ( e ) ∗ . The q -von Neumann algebra Γ q ( H ) is the von Neumann algebra generated by the operators ω ( e ) where e ∈ H . It is a finite vonNeumann algebra with the trace τ defined by τ ( x ) = h Ω , x. Ω i F q ( H ) where x ∈ Γ q ( H ) .Let H and K be real Hilbert spaces and H T −→ K be a contraction with complexification H C T C −→ K C .We define the following linear map F q ( T ) : F q ( H ) −→ F q ( K ) h ⊗ · · · ⊗ h n T C h ⊗ · · · ⊗ T C h n . Then there exists a unique map Γ q ( H ) Γ q ( T ) −−−−→ Γ q ( H ) such that for every x ∈ Γ q ( H ) we have (cid:0) Γ q ( T )( x ) (cid:1) Ω = F q ( T )( x Ω) . This map is normal, unital, completely positive and trace preserving. If H T −→ K is an isometry, Γ q ( T ) is an injective ∗ -homomorphism. If p < ∞ , it extends to a contraction L p (cid:0) Γ q ( H ) (cid:1) Γ q ( T ) −−−−→ L p (cid:0) Γ q ( K ) (cid:1) .We are mainly concerned with the fermion algebra Γ − ( H ) . In this case, recall that if e ∈ H hasnorm 1, then the operator ω ( e ) satisfies ω ( e ) = Id F − ( H ) . Moreover, we need the following Wickformula, (see [Boz, page 2] and [EfP, Corollary 2.1]). In order to state this, we denote, if k > isan integer, by P (2 k ) the set of 2-partitions of the set { , , . . . , k } . If V ∈ P (2 k ) we let c ( V ) thenumber of crossings of V , which is given, by the number of pairs of blocks of V which cross (see [EfP,page 8630] for a precise definition). Then, if f , . . . , f k ∈ H we have τ (cid:0) ω ( f ) ω ( f ) · · · ω ( f k ) (cid:1) = X V∈P (2 k ) ( − c ( V ) Y ( i,j ) ∈V h f i , f j i H . (4.2)In particular, for all e, f ∈ H , we have τ (cid:0) ω ( e ) ω ( f ) (cid:1) = h e, f i H . (4.3)Let A = [ a ij ] i,j ∈ I be a matrix of M I . By definition, the Schur multiplier on B (cid:0) ℓ I (cid:1) associatedwith this matrix is the unbounded linear operator M A whose domain is the space of all B = [ b ij ] i,j ∈ I of B (cid:0) ℓ I (cid:1) such that [ a ij b ij ] i,j ∈ I belongs to B (cid:0) ℓ I (cid:1) , and whose action on B = [ b ij ] i,j ∈ I is given by M A ( B ) = [ a ij b ij ] i,j ∈ I . For all i, j ∈ I , the matrix e ij belongs to D ( M A ) , hence M A is densely definedfor the weak* topology. Suppose p < ∞ . If for any B ∈ S pI , we have B ∈ D ( M A ) and thematrix M A ( B ) represents an element of S pI , by the closed graph theorem, the matrix A of M I definesa bounded Schur multiplier S pI M A −−→ S pI . We have a similar statement for bounded Schur multiplierson B (cid:0) ℓ I (cid:1) .Recall that a matrix A of M I defines a contractive Schur multiplier B (cid:0) ℓ I (cid:1) M A −−→ B (cid:0) ℓ I (cid:1) if and only ifthere exists an index set K and norm 1 vectors h i ∈ ℓ K and k j ∈ ℓ K such that for all i, j ∈ I we have a i,j = h h i , k j i ℓ K (see [Pau]). If all entries of A are real numbers, we can take the real vector space ℓ K ( R ) instead of the complex vector space ℓ K . Finally, recall that every contractive Schur multiplier B (cid:0) ℓ I (cid:1) M A −−→ B (cid:0) ℓ I (cid:1) is completely contractive (see [Pau]).We say that a matrix A of M I induces a completely positive Schur multiplier B (cid:0) ℓ I (cid:1) M A −−→ B (cid:0) ℓ I (cid:1) ifand only if for any finite set F ⊂ I the matrix [ a i,j ] i,j ∈ F is positive (see [Pau]). An other well-knowncharacterization is that there exists vectors h i ∈ ℓ K ( C ) of norm 1 such that for all i, j ∈ I we have a i,j = h h i , h j i ℓ K . If A is a real matrix, we can use the real vector space ℓ K ( R ) instead of the complexvector space ℓ K . 13et M be a von Neumann algebra equipped with a semifinite normal faithful trace τ . Supposethat M T −→ M is a normal contraction. We say that T is selfadjoint if for all x, y ∈ M ∩ L ( M ) wehave τ (cid:0) T ( x ) y ∗ (cid:1) = τ (cid:0) x ( T y ) ∗ (cid:1) . In this case, it is easy to see that the restriction T | M ∩ L ( M ) extends to a contraction L ( M ) T −→ L ( M ) . By complex interpolation, for any p ∞ , we obtain a contractive map L p ( M ) T −→ L p ( M ) .Moreover, the operator L ( M ) T −→ L ( M ) is selfadjoint. If M T −→ M is a normal selfadjoint completecontraction, it is easy to see that the map L p ( M ) T −→ L p ( M ) is completely contractive for all p ∞ . It is easy to see that a contractive Schur multiplier B (cid:0) ℓ I (cid:1) M A −−→ B (cid:0) ℓ I (cid:1) associated with a matrix A of M I is selfadjoint if and only if all entries of A are real.In order to prove the next theorem, we need the following notion of infinite tensor product of vonNeumann algebras, see [Tak3]. Given a sequence ( M n , τ n ) n ∈ Z of von Neumann algebras M n equippedwith faithful normal finite traces τ n , then on the infinite minimal C ∗ -tensor product of the algebras ( M n ) n ∈ Z there is a well-defined infinite product state · · · ⊗ τ − ⊗ τ ⊗ τ ⊗ · · · . The weak operatorclosure of the GNS-representation of the infinite C ∗ -tensor product of ( M n ) n ∈ Z with respect to thestate · · · ⊗ τ − ⊗ τ ⊗ τ ⊗ · · · yields a von Neumann algebra, called the infinite tensor product of vonNeumann algebras M n with respect to the traces τ n . We will denote this algebra by N n ∈ Z ( M n , τ n ) .The state · · · ⊗ τ − ⊗ τ ⊗ τ ⊗ · · · extends to a faithful normal finite trace on N n ∈ Z ( M n , τ n ) whichwe still denote by · · · ⊗ τ − ⊗ τ ⊗ τ ⊗ · · · .The following theorem states that we can dilate some Schur multipliers. The construction (andthe one of Theorem 4.6) is inspired by the work of É. Ricard [Ric]. Theorem 4.2
Let B (cid:0) ℓ I (cid:1) M A −−→ B (cid:0) ℓ I (cid:1) be a unital completely positive Schur multiplier associated witha real-valued matrix A . Then there exists a hyperfinite von Neumann algebra M equipped with asemifinite normal faithful trace, a unital trace preserving ∗ -automorphism M U −→ M , a unital tracepreserving one-to-one normal ∗ -homomorphism B (cid:0) ℓ I (cid:1) J −→ M such that ( M A ) k = E U k J for any integer k > , where M E −→ B (cid:0) ℓ I (cid:1) is the canonical faithful normal trace preserving conditionalexpectation associated with J .Proof : Since the map B (cid:0) ℓ I (cid:1) M A −−→ B (cid:0) ℓ I (cid:1) is completely positive we can define a positive symmetricbilinear form h· , ·i ℓ ,A on the real span of the e i , where i ∈ I , by: h e i , e j i ℓ ,A = a ij . (4.4)We denote by ℓ ,A the completion of the real pre-Hilbert obtained by quotient by the correspondingkernel. For all i of I we still denote by e i the class of e i in ℓ ,A . Now we define the von Neumannalgebra M by M = B (cid:0) ℓ I (cid:1) ⊗ O n ∈ Z (cid:0) Γ − ( ℓ ,A ) , τ (cid:1)! . Since the von Neumann algebra Γ − ( ℓ ,A ) is hyperfinite, the von Neumann algebra M is also hyper-finite. We define the element d of M by d = X i ∈ I e ii ⊗ · · · ⊗ I ⊗ ω ( e i ) ⊗ I ⊗ · · · ω ( e i ) is in position 0. Recall that M A is unital. Then it is not difficult to see that d is asymmetry, i.e. a selfadjoint unitary element. We equip the von Neumann algebra M with the faithfulsemifinite normal trace τ M = Tr ⊗ · · · ⊗ τ ⊗ τ ⊗ · · · . We denote by M E −→ B (cid:0) ℓ I (cid:1) the canonical faithfulnormal trace preserving conditional expectation of M onto B (cid:0) ℓ I (cid:1) . We have E = Id B ( ℓ I ) ⊗ · · · ⊗ τ ⊗ τ ⊗ · · · . We define the canonical injective normal unital ∗ -homomorphism J : B (cid:0) ℓ I (cid:1) −→ Mx x ⊗ · · · ⊗ I ⊗ I ⊗ · · · . Clearly, J preserves the traces. We define the right shift S : N n ∈ Z (cid:0) Γ − ( ℓ ,A ) , τ (cid:1) −→ N n ∈ Z (cid:0) Γ − ( ℓ ,A ) , τ (cid:1) · · · ⊗ x ⊗ x ⊗ · · · 7−→ · · · ⊗ x − ⊗ x ⊗ · · · . Now, we define the linear map U : M −→ My d (cid:0) ( Id B ( ℓ I ) ⊗ S )( y ) (cid:1) d. The map M U −→ M is a unital ∗ -automorphism of M . Moreover, it is easy to see that M U −→ M preserves the trace τ M . Now, we will show that, for any positive integer k , we have, for all x ∈ B (cid:0) ℓ I (cid:1) U k ◦ J ( x ) = X i,j ∈ I x ij e ij ⊗ · · · ⊗ I ⊗ ω ( e i ) ω ( e j ) ⊗ · · · ⊗ ω ( e i ) ω ( e j ) | {z } k factors ⊗ I ⊗ · · · (4.5)by induction on k , where the first ω ( e i ) ω ( e j ) is in position . The statement clearly holds for k = 0 .Now, assume (4.5). For all x ∈ B (cid:0) ℓ I (cid:1) , we have U k +1 ◦ J ( x ) = d (cid:16) ( Id B ( ℓ I ) ⊗ S ) (cid:0) U k ◦ J ( x ) (cid:1)(cid:17) d = d ( Id B ( ℓ I ) ⊗ S ) X i,j ∈ I x ij e ij ⊗ · · · ⊗ I ⊗ ω ( e i ) ω ( e j ) ⊗ · · · ⊗ ω ( e i ) ω ( e j ) ⊗ I ⊗ · · · !! d = X r ∈ I e rr ⊗ · · · ⊗ I ⊗ ω ( e r ) ⊗ I ⊗ · · · ! X i,j ∈ I x ij e ij ⊗ · · · ⊗ I ⊗ I ⊗ ω ( e i ) ω ( e j ) ⊗ · · ·⊗ ω ( e i ) ω ( e j ) ⊗ I ⊗ · · · ! X s ∈ I e ss ⊗ · · · ⊗ I ⊗ ω ( e s ) ⊗ I ⊗ · · · ! = X i,j,r,s ∈ I x ij e rr e ij e ss ⊗ · · · ⊗ I ⊗ ω ( e r ) ω ( e s ) ⊗ ω ( e i ) ω ( e j ) ⊗ · · · ⊗ ω ( e i ) ω ( e j ) ⊗ I ⊗ · · · = X i,j ∈ I x ij e ij ⊗ · · · ⊗ I ⊗ ω ( e i ) ω ( e j ) ⊗ ω ( e i ) ω ( e j ) ⊗ · · · ⊗ ω ( e i ) ω ( e j ) ⊗ I ⊗ · · · .
15e obtained the statement (4.5) for k + 1 . Then, we deduce that for any positive integer k and any x ∈ B (cid:0) ℓ I (cid:1) we have E ◦ U k ◦ J ( x )= ( Id S pI ⊗ · · · ⊗ τ ⊗ · · · ) X i,j ∈ I x ij e ij ⊗ · · · ⊗ I ⊗ ω ( e i ) ω ( e j ) ⊗ · · · ⊗ ω ( e i ) ω ( e j ) ⊗ I ⊗ · · · ! = X i,j ∈ I τ (cid:0) ω ( e i ) ω ( e j ) (cid:1) k x ij e ij = X i,j ∈ I (cid:0) h e i , e j i ℓ ,A (cid:1) k x ij e ij by (4.3) = X i,j ∈ I ( a ij ) k x ij e ij by (4.4) = ( M A ) k ( x ) . Thus, for any positive integer k , we have ( M A ) k = E ◦ U k ◦ J. The proof is complete.
Corollary 4.3
Let B (cid:0) ℓ I (cid:1) M A −−→ B (cid:0) ℓ I (cid:1) be a contractive Schur multiplier associated with a real-valuedmatrix A . Suppose < p < ∞ . Then, the induced Schur multiplier S pI M A −−→ S pI satisfies the noncom-mutative Matsaev’s property. More precisely, for any complex polynomial P , we have (cid:13)(cid:13) P ( M A ) (cid:13)(cid:13) cb,S pI −→ S pI k P k p,S p . Proof : Suppose that B (cid:0) ℓ I (cid:1) M A −−→ B (cid:0) ℓ I (cid:1) is a contractive Schur multiplier associated with a real matrix A of M I . There exists a set K and norm 1 vectors h i ∈ ℓ K ( R ) and k i ∈ ℓ K ( R ) such that for all i, j ∈ I we have a i,j = h h i , k j i ℓ K ( R ) . Now we define the following matrices of M I B = h h h i , h j i ℓ K ( R ) i i,j ∈ I , C = h h k i , k j i ℓ K ( R ) i i,j ∈ I and D = h h k i , h j i ℓ K ( R ) i i,j ∈ I . For all i ∈ I and all n ∈ { , } , we define the norm 1 vector l ( n,i ) of ℓ K ( R ) by l ( n,i ) = (cid:26) h i if n = 1 and i ∈ Ik i if n = 2 and i ∈ I. Now, by the identification M ( M I ) ≃ M { , }× I , the matrix (cid:20) B AD C (cid:21) of M ( M I ) identifies to thematrix F = h h l n,i , l m,j i ℓ K ( R ) i ( n,i ) ∈{ , }× I, ( m,j ) ∈{ , }× I of M { , }× I . The Schur multiplier B (cid:0) ℓ { , }× I (cid:1) M F −−→ B (cid:0) ℓ { , }× I (cid:1) associated with this matrix is uni-tal and completely positive. Moreover, since the matrix F is real, B (cid:0) ℓ { , }× I (cid:1) M F −−→ B (cid:0) ℓ { , }× I (cid:1) is16elfadjoint. Let < p < ∞ . For any complex polynomial P , we have (cid:13)(cid:13) P ( M A ) (cid:13)(cid:13) S pI −→ S pI (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:20) P ( M B ) P ( M A ) P ( M D ) P ( M C ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S p ( S pI ) −→ S p ( S pI ) = (cid:13)(cid:13) P ( M F ) (cid:13)(cid:13) S p { , }× I −→ S p { , }× I . Now, according to Theorem 4.2, remarks following Lemma 4.1 and Corollary 2.6, the Schur multiplier M F satisfies the noncommutative Matsaev’s property. We deduce that M A also satisfies this property.Moreover, in applying this result to the Schur multiplier M I ⊗ A (= I ⊗ M A ) , we obtain the inequalityfor the completely bounded norm.We pass to Fourier multipliers on discrete groups. Suppose that G is a discrete group. We denoteby e G the neutral element of G . We denote by ℓ G λ ( g ) −−−→ ℓ G the unitary operator of left translation by g and VN( G ) the von Neumann algebra of G spanned by the λ ( g ) where g ∈ G . It is an finite algebrawith trace given by τ G ( x ) = (cid:10) ε e G , x ( ε e G ) (cid:11) ℓ G where ( ε g ) g ∈ G is the canonical basis of ℓ G and x ∈ VN( G ) . A Fourier multiplier is a normal linearmap VN( G ) T −→ VN( G ) such that there exists a function G t −→ C such that for all g ∈ G we have T (cid:0) ( λ ( g ) (cid:1) = t g λ ( g ) . In this case, we denote T by M t : VN( G ) −→ VN( G ) λ ( g ) t g λ ( g ) . It is easy to see that a contractive Fourier multiplier
VN( G ) M t −−→ VN( G ) is selfadjoint if and only if G t −→ C is a real function. It is well-known that a Fourier multiplier VN( G ) M t −−→ VN( G ) is completelypositive if and only if the function t is positive definite. If the discrete group G is amenable, by [DCH,Corollary 1.8], every contractive Fourier multiplier VN( G ) M t −−→ VN( G ) is completely contractive.Recall the following particular case of the transfer result [NeR, Theorem 2.7]. Theorem 4.4
Let G be an amenable discrete group. Suppose < p < ∞ . Let G t −→ R a function.Let A be the matrix of M G defined by a g,h = t gh − where g, h ∈ G . The Fourier multiplier M t iscompletely bounded on L p (VN( G )) if and only if the Schur multiplier M A is completely bounded on S pG . In this case, we have (4.6) (cid:13)(cid:13) M t (cid:13)(cid:13) cb,L p (VN( G )) −→ L p (VN( G )) = (cid:13)(cid:13) M A (cid:13)(cid:13) cb,S pG −→ S pG . Now, we can prove the next result.
Corollary 4.5
Let G be an amenable discrete group. Let VN( G ) M t −−→ VN( G ) be a contractive Fouriermultiplier associated with a real function G t −→ R . Suppose < p < ∞ . Then, the induced Fourier mul-tiplier L p (cid:0) VN( G ) (cid:1) M t −−→ L p (cid:0) VN( G ) (cid:1) satisfies the noncommutative Matsaev’s property. More precisely,for any complex polynomial P , we have (cid:13)(cid:13) P ( M t ) (cid:13)(cid:13) cb,L p (VN( G )) −→ L p (VN( G )) k P k p,S p . Proof : We define the matrix A of M G by a g,h = t gh − where g, h ∈ G . By (4.6), for any complexpolynomial P and all < p < ∞ , we have (cid:13)(cid:13) P ( M t ) (cid:13)(cid:13) cb,L p (VN( G )) −→ L p (VN( G )) = (cid:13)(cid:13) P ( M A ) (cid:13)(cid:13) cb,S pG −→ S pG = (cid:13)(cid:13) P ( Id S p ⊗ M A ) (cid:13)(cid:13) S p ( S pG ) −→ S p ( S pG ) = (cid:13)(cid:13) P ( M I ⊗ A ) (cid:13)(cid:13) S p ( S pG ) −→ S p ( S pG ) . G is amenable, the Fourier multiplier VN( G ) M t −−→ VN( G ) is completely contractive. Moreover,the map VN( G ) M t −−→ VN( G ) is selfadjoint. Thus, for any < p < ∞ , the map L p (cid:0) VN( G ) (cid:1) M t −−→ L p (cid:0) VN( G ) (cid:1) is completely contractive. Then, by (4.6), for any < p < ∞ , we have k M I ⊗ A k S p ( S pG ) −→ S p ( S pG ) = k M A k cb,S pG −→ S pG = k M t k cb,L p (VN( G )) −→ L p (VN( G )) . By Corollary 4.3, we deduce finally that, for any complex polynomial P and all < p < ∞ , we have (cid:13)(cid:13) P ( M t ) (cid:13)(cid:13) L p (VN( G )) −→ L p (VN( G )) k P k p,S p ( S pG ) = k P k p,S p . In order to prove the next theorem we need the notion of crossproduct. We refer to [Str] and [Sun]for more information on this concept.
Theorem 4.6
Let G be a discrete group. Let VN( G ) M t −−→ VN( G ) be a unital completely positiveFourier multiplier associated with a real valued function G t −→ R . Then there exists a von Neumannalgebra M equipped with a faithful finite normal trace, a unital trace preserving ∗ -automorphism M U −→ M , a unital normal trace preserving one-to-one ∗ -homomorphism VN( G ) J −→ M such that, ( M t ) k = E U k J for any integer k > , where M E −→ V N ( G ) is the canonical faithful normal trace preserving conditionalexpectation associated with J .Proof : Since the map VN( G ) M t −−→ VN( G ) is completely positive, we can define a positive symmetricbilinear form h· , ·i ℓ ,t on the real span of the e g , where g ∈ G , by: h e g , e h i ℓ ,t = t g − h . We denote by ℓ ,t the completion of the real pre-Hilbert space obtained by quotient by the corre-sponding kernel. For all g ∈ G , we denote by g the class of e g in ℓ ,t . Then, for all g, h ∈ G , wehave h g, h i ℓ ,t = t g − h . For all g ∈ G , it easy to see that there exists a unique isometry ℓ ,t θ g −→ ℓ ,t such that for all h ∈ G wehave θ g ( h ) = gh. For all g ∈ G , we define the unital trace preserving ∗ -automorphism α ( g ) = Γ − ( θ g ⊗ Id ℓ Z ) : α ( g ) : Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) −→ Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) w ( h ⊗ v ) w ( gh ⊗ v ) . The homomorphism G α −→ Aut (cid:16) Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1)(cid:17) allows us to define the von Neumann algebra M = Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) ⋊ α G. (4.7)We can identify Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) as a subalgebra of M . We let J the canonical normal unital injective ∗ -homomorphism VN( G ) J −→ M . We denote by τ the faithful finite normal trace on Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) .18ecall that, for all g ∈ G , the map α ( g ) is trace preserving. Thus, the trace τ is α -invariant. Weequip M with the induced canonical trace τ M . For all x ∈ Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) and all g ∈ G , we have(4.8) τ M (cid:16) xJ (cid:0) λ ( g ) (cid:1)(cid:17) = δ g,e G τ ( x ) (see [Str] pages 359 and 352). If g, h ∈ G and v ∈ ℓ Z , we can write the relations of commutation ofthe crossed product as(4.9) J (cid:0) λ ( g ) (cid:1) ω ( h ⊗ v ) = ω ( gh ⊗ v ) J (cid:0) λ ( g ) (cid:1) . We denote by M E −→ VN( G ) the canonical faithful normal trace preserving conditional expectation.For all x ∈ Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) and all g ∈ G we have E (cid:16) xJ (cid:0) λ ( g ) (cid:1)(cid:17) = τ ( x ) λ ( g ) . We define the unital trace preserving ∗ -automorphism S = Γ − ( Id ℓ ,t ⊗ S ) : S : Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) −→ Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) ω ( h ⊗ e n ) ω ( h ⊗ e n +1 ) . Since M t is unital, ω ( e G ⊗ e ) is a symmetry, i.e. a selfadjoint unitary element. Moreover, for all g ∈ G ,we have α ( g ) S = S α ( g ) . Then, by [Sun, Proposition 4.4.4], we can define a unital ∗ -automorphism U : M −→ MxJ (cid:0) λ ( g ) (cid:1) ω ( e G ⊗ e ) S ( x ) J (cid:0) λ ( g ) (cid:1) ω ( e G ⊗ e ) . Now, we will show that U preserves the trace. For all g ∈ G and all x ∈ Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) , we have τ M (cid:18) U (cid:16) xJ (cid:0) λ ( g ) (cid:1)(cid:17)(cid:19) = τ M (cid:16) S ( x ) J (cid:0) λ ( g ) (cid:1)(cid:17) = δ g,e G τ (cid:0) S ( x ) (cid:1) by (4.8) = δ g,e G τ ( x )= τ M (cid:16) xJ (cid:0) λ ( g ) (cid:1)(cid:17) . We conclude by linearity and normality. It is not hard to see that J preserves the traces.Now, we will prove that, for any integer k > and any g ∈ G , we have U k ◦ J (cid:0) λ ( g ) (cid:1) (4.10) = ω ( e G ⊗ e ) ω ( e G ⊗ e ) · · · ω ( e G ⊗ e k − ) ω ( g ⊗ e k − ) ω ( g ⊗ e k − ) · · · ω ( g ⊗ e ) J (cid:0) λ ( g ) (cid:1) by induction on k . The statement holds clearly for k = 1 : if g ∈ G , we have U ◦ J (cid:0) λ ( g ) (cid:1) = ω ( e G ⊗ e ) J (cid:0) λ ( g ) (cid:1) ω ( e G ⊗ e )= ω ( e G ⊗ e ) ω ( g ⊗ e ) J (cid:0) λ ( g ) (cid:1) by (4.9) . Now, assume (4.10). For all g ∈ G , we have U k +1 ◦ J (cid:0) λ ( g ) (cid:1) = U (cid:16) ω ( e G ⊗ e ) ω ( e G ⊗ e ) · · · ω ( e G ⊗ e k − ) ω ( g ⊗ e k − ) ω ( g ⊗ e k − ) · · · ω ( g ⊗ e ) J (cid:0) λ ( g ) (cid:1)(cid:17) = ω ( e G ⊗ e ) ω ( e G ⊗ e ) ω ( e G ⊗ e ) · · · ω ( e G ⊗ e k ) ω ( g ⊗ e k ) ω ( g ⊗ e k − ) · · · ω ( g ⊗ e ) J (cid:0) λ ( g ) (cid:1) ω ( e G ⊗ e )= ω ( e G ⊗ e ) ω ( e G ⊗ e ) ω ( e G ⊗ e ) · · · ω ( e G ⊗ e k ) ω ( g ⊗ e k ) ω ( g ⊗ e k − ) · · · ω ( g ⊗ e ) ω ( g ⊗ e ) J (cid:0) λ ( g ) (cid:1) .
19e obtained the statement (4.10) for k + 1 . Now let k > and g ∈ G . We define the elements f , . . . , f k of the Hilbert space ℓ ,t ⊗ ℓ Z by f i = (cid:26) e G ⊗ e i − if i kg ⊗ e k − i if k + 1 i k If i k , we have h f i , f k − i +1 i ℓ ,t ⊗ ℓ Z = h e G ⊗ e i − , g ⊗ e i − i ℓ ,t ⊗ ℓ Z = h e G , g i ℓ ,t h e i − , e i − i ℓ Z = t g By a similar computation, if i < j k with j = 2 k − i + 1 , we obtain h f i , f j i ℓ ,t ⊗ ℓ Z = 0 . Then,for all g ∈ G , we have E U k J (cid:0) λ ( g ) (cid:1) = E (cid:16) ω ( e G ⊗ e ) · · · ω ( e G ⊗ e k − ) ω ( g ⊗ e k − ) · · · ω ( g ⊗ e ) J (cid:0) λ ( g ) (cid:1)(cid:17) = τ (cid:0) ω ( e G ⊗ e ) · · · ω ( e G ⊗ e k − ) ω ( g ⊗ e k − ) · · · ω ( g ⊗ e ) (cid:1) λ ( g )= τ (cid:0) ω ( f ) ω ( f ) · · · ω ( f k ) (cid:1) λ ( g )= X V∈P (2 k ) ( − c ( V ) Y ( i,j ) ∈V h f i , f j i ℓ ,t ⊗ ℓ Z ! λ ( g ) by (4.2)(4.11) = h f , f k i ℓ ,t ⊗ ℓ Z · · · h f k , f k +1 i ℓ ,t ⊗ ℓ Z λ ( g )= ( t g ) k λ ( g )= T k (cid:0) λ ( g ) (cid:1) . (cid:0) The only non-zero term in the sum of (4.11) is the term with V = (cid:8) (1 , k ) , (2 , k − , . . . , ( k, k + 1) (cid:9) ,which satisfies c ( V ) = 0 (cid:1) . Thus, for any positive integer k (the case k = 0 is trivial), we conclude that T k = E U k J. Corollary 4.7
Let G be a discrete group. Let VN( G ) M t −−→ VN( G ) be a unital completely positiveFourier multiplier which is associated with a real function G t −→ R . Suppose that the von Neumannalgebra (4.7) has QWEP. Let p ∞ . Then, the induced Fourier multiplier L p (cid:0) VN( G ) (cid:1) M t −−→ L p (cid:0) VN( G ) (cid:1) satisfies the noncommutative Matsaev’s property.Proof : This corollary follows from Theorem 4.6, remarks following Lemma 4.1 and Corollary 2.6.At the light of above corollary, it is important to know when the von Neumann algebra (4.7)has QWEP. If the group G is amenable, this algebra has QWEP by [Oza, Proposition 4.1] (or [Con,Proposition 6.8]). Now we give an example of non-amenable group G such that this von Neumannalgebra has QWEP. We denote by F n a free group with n generators denoted by g , . . . , g n where n ∞ . We denote by R the hyperfinite factor of type II and by R U an ultrapower of R with respect to a non-trivial ultrafilter U . In order to prove the next theorem we need the notion ofamalgamated free product of von Neumann algebras. We refer to [BlD] and [Ued] for more informationon this concept. Note that, with the notations of the proof of Theorem 4.6, the von Neumann algebra Γ − (cid:0) ℓ ,t ⊗ ℓ Z (cid:1) is ∗ -isomorphic to the hyperfinite factor of type II .20 roposition 4.8 Suppose n ∞ . Let F n α −→ Aut( R ) be a homomorphism. Then the crossedproduct R ⋊ α F n has QWEP.Proof : First we will show the result for n = 2 . We denote by h g i and h g i the subgroups of F generated by g and g and by α and α the restrictions of α to these subgroups. First, we provethat the subalgebras R ⋊ α h g i and R ⋊ α h g i are free with respect to the canonical faithful normaltrace preserving conditional expectation R ⋊ α F E −→ R . We identify R as a subalgebra of R ⋊ α F .We may regard the elements of R ⋊ α F as matrices h α r − (cid:0) ̟ ( rt − ) (cid:1)i r,t ∈ F with entries in R where F ̟ −→ R is a map. Recall that the conditional expectation E on R is given by E (cid:18)h α r − (cid:0) ̟ ( rt − ) (cid:1)i r,t ∈ F (cid:19) = ̟ ( e F ) . Suppose that i , . . . , i k ∈ { , } are integers such that i = i , . . . , i k − = i k . For any j k , let A j = h α r − (cid:0) ̟ j ( rt − ) (cid:1)i r,t ∈ F be an element of R ⋊ α ij h g i j i such that E ( A j ) = 0 where each F ̟ j −−→ R is a map satisfying ̟ j ( g ) = 0 if g
6∈ h g i j i . Then, for all j k , we have ̟ j ( e F ) = 0 . Now, we have E ( A · · · A k ) = E (cid:18)h α r − (cid:0) ̟ ( rt − ) (cid:1)i r,t ∈ F · · · h α r − (cid:0) ̟ k ( rt − ) (cid:1)i r,t ∈ F (cid:19) = X L ,...,l k − ∈ F ̟ (cid:0) l − (cid:1) α l − (cid:16) ̟ (cid:0) l l − (cid:1)(cid:17) · · · α l − k − (cid:16) ̟ (cid:0) l k − l − k − (cid:1)(cid:17) α l − k − (cid:0) ̟ k ( l k − ) (cid:1) = 0 . Thus the von Neumann algebra R ⋊ α F decomposes as an amalgamated free product of R ⋊ α h g i and R ⋊ α h g i over R . Moreover, the groups h g i and h g i are commutative, hence amenable. Wehave already point out that the crossed product of the hyperfinite factor R by an amenable grouphas QWEP. Then the von Neumann algebras R ⋊ α h g i and R ⋊ α h g i are QWEP. Moreover, by[Bla, page 283], these von Neumann algebras have a separable predual. By [Kir, Theorem 1.4], wededuce that these von Neumann algebras are embeddable into R U . Now, the theorem stating in [BDJ,Corollary 4.5] says that, for finite von Neumann algebras with separable preduals, being embeddableinto R U is stable under amalgamated free products over a hyperfinite von Neumann algebra. Thuswe deduce that R ⋊ α F is embeddable into R U , which is equivalent to QWEP, by [Kir, Theorem 1.4],since R ⋊ α F has a separable predual. Induction then gives the case when n < ∞ , and the case n = ∞ then follows since, by [Oza, Proposition 4.1], QWEP is preserved by taking the weak* closureof increasing unions of von Neumann algebras.We pass to maps arising in the second quantization in the context of [BKS]. Proposition 4.9
Suppose < p < ∞ and − q < . Let H be a real Hilbert space and H T −→ H a contraction. Then the induced map L p (cid:0) Γ q ( H ) (cid:1) M t −−→ L p (cid:0) Γ q ( H ) (cid:1) satisfies the noncommutativeMatsaev’s property.Proof : There exists an orthogonal dilation K U −→ K de H T −→ H . We denote by H J −→ K theembedding of H in K and K Q −→ H the projection of K on H . The map Γ q ( K ) Γ( J ) −−−→ Γ q ( K ) is aunital injective normal trace preserving ∗ -homomorphism. The map Γ q ( H ) Γ( U ) −−−→ Γ q ( K ) is a unital21race preserving ∗ -automorphism. The map Γ q ( K ) Γ( Q ) −−−→ Γ q ( H ) is the canonical faithful normal unitaltrace preserving conditional expectation of Γ q ( K ) on Γ q ( H ) . Moreover, we have for any integer k Γ q ( T ) k = Γ q ( P )Γ q ( U ) k Γ q ( J ) . We conclude with Theorem 4.6, remarks following Lemma 4.1, Corollary 2.6 and by using the factthat, by [Nou], the von Neumann algebra Γ q ( H ) has QWEP.In order to state more easily our following result we need to define the following property. Let M be a von Neumann algebra. Suppose that M T −→ M is a linear map. Property 4.10
There exists a von Neumann algebra N with QWEP equipped with a normal faithfulfinite trace on N , a unital trace preserving ∗ -automorphism N U −→ N , a unital injective normal tracepreserving ∗ -homomorphism M J −→ N such that, T k = E U k J. for any integer k > , where M E −→ V N ( G ) is the canonical faithful normal trace preserving conditionalexpectation associated with J . This property is stable under free product. Indeed, one can prove the next proposition with anargument similar to that used in the proof of [JMX, Lemma 10.4] and by using [BDJ, Corollary 4.5]and [Kir, Theorem 1.4].
Proposition 4.11
Let M and M be von Neumann algebras with separable preduals equipped withnormal faithful finite traces τ and τ . Let M T −→ M and M T −→ M be linear maps. If T and T satisfy Property 4.10, their free product ( M , τ ) ∗ ( M , τ ) T ∗ T −−−−→ ( M , τ ) ∗ ( M , τ ) also satisfies Property 4.10. Thus the above proposition allows us to construct other examples of contractions satisfying the non-commutative Matsaev’s property.
Suppose p < ∞ . We denote by ( T t ) t > the translation semigroup on L p ( R ) , where T t ( f )( s ) = f ( s − t ) if f ∈ L p ( R ) and s, t ∈ R . This semigroup ( T t ) t > is a C -semigroup of contractions.Let ( T t ) t > be a C -semigroup of contractions on a Banach space X . For all b ∈ L ( R ) withsupport in R + , it is easy to see that the linear operator R + ∞ b ( t ) T t dt : X −→ Xx R + ∞ b ( t ) T t xdt is well-defined and bounded. Moreover, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X −→ X k b k L ( R ) . Now, let us state a question for semigroups which is analogue to Matsaev’s Conjecture 1.1.22 uestion 5.1
Suppose < p < ∞ , p = 2 . Let ( T t ) t > be a C -semigroup of contractions on a L p -space L p (Ω) of a measure space Ω . Do we have the following estimate (5.1) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) −→ L p (Ω) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R ) −→ L p ( R ) for all b ∈ L ( R ) with support in R + ? We pass to the noncommutative case. We can state the following noncommutative analogue ofQuestion 5.1.
Question 5.2
Suppose < p < ∞ , p = 2 . Let ( T t ) t > be a C -semigroup of contractions on anoncommutative L p -space L p ( M ) . Do we have the following estimate (5.2) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( M ) −→ L p ( M ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) cb,L p ( R ) −→ L p ( R ) for all b ∈ L ( R ) with support in R + ? For all b ∈ L ( R ) with support in R + , it is clear that C b = R + ∞ b ( t ) T t dt . Moreover, for all b ∈ L ( R ) , we have k C b k L ( R ) −→ L ( R ) = k C b k cb,L ( R ) −→ L ( R ) = k b k L ( R ) . Consequently, the inequalities (5.1) and (5.2) hold true for p = 1 .In [CoW, page 25], it is proved that the C -semigroups of positive contractions satisfy inequality(5.1). Using [Pel1, Theorem 3] and the same method, we can generalize this result to C -semigroupsof operators which admit a contractive majorant. Now, we adapt this method in order to give a linkbetween Question 5.2 and Question 1.3. Theorem 5.3
Suppose < p < ∞ . Let ( T t ) t > be a C -semigroup of contractions on a noncom-mutative L p -space L p ( M ) such that each L p ( M ) T t −→ L p ( M ) satisfies the noncommutative Matsaev’sproperty. Then the semigroup ( T t ) t > satisfies inequality (5.2).Proof : It is not hard to see that it suffices to prove this in the case when b has compact support.Now we define the sequence (cid:0) a n (cid:1) n > of complex sequences indexed by Z as in the proof of Theorem3.5. Let n > . Observe that if R + f −→ L p ( M ) is continuous and piecewise affine with nodes at kn then Z + ∞ b ( t ) f ( t ) dt = + ∞ X k =0 a n,k f (cid:18) kn (cid:19) . Let x ∈ L p ( M ) . Let R + f n −→ L p ( M ) be the continuous and piecewise affine function with nodes at kn such that f n (cid:0) kn (cid:1) = (cid:0) T n (cid:1) k x . Since the map t T t x is uniformly continuous on compacts of R + wehave (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t xdt − + ∞ X k =0 a n,k (cid:0) T n (cid:1) k x (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( M ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t xdt − + ∞ X k =0 a n,k f n (cid:18) kn (cid:19)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( M ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) (cid:0) T t x − f n ( t ) (cid:1) dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( M ) −−−−−→ n → + ∞ .
23e deduce that + ∞ X k =0 a n,k (cid:0) T n (cid:1) k so −−−−−→ n → + ∞ Z + ∞ b ( t ) T t dt. By the commutative diagram of the proof of Theorem 3.5, we have for any integer n > (cid:13)(cid:13) C a n (cid:13)(cid:13) cb,ℓ p Z −→ ℓ p Z k C b k cb,L p ( R ) −→ L p ( R ) . Finally, by the strongly lower semicontinuity of the norm, we obtain that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( M ) −→ L p ( M ) lim inf n → + ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + ∞ X k =1 a n,k (cid:0) T n (cid:1) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( M ) −→ L p ( M ) lim inf n → + ∞ k C a n k cb,ℓ p Z −→ ℓ p Z = k C b k cb,L p ( R ) −→ L p ( R ) . The first consequence of this theorem is that inequality (5.2) holds true for p = 2 . Now, we listsome natural examples of semigroups which satisfy inequality (5.2) by our results, using this theorem. Semigroups of Schur multipliers.
Let ( T t ) t > a w*-semigroup of selfadjoint contractive Schurmultipliers on B (cid:0) ℓ I (cid:1) . If p < ∞ and t > , the map B (cid:0) ℓ I (cid:1) T t −→ B (cid:0) ℓ I (cid:1) induces a contraction S pI T t −→ S pI . Using [JMX, Remark 5.2], it is easy to see that we obtain a C -semigroup of contractions S pI T t −→ S pI which satisfies inequality (5.2). Semigroups of Fourier multipliers on an amenable group.
Let G be an amenable group. Let ( T t ) t > a w*-semigroup of selfadjoint contractive Fourier multipliers on V N ( G ) . If p < ∞ and t > , the map V N ( G ) T t −→ V N ( G ) induces a contraction L p (cid:0) V N ( G ) (cid:1) T t −→ L p (cid:0) V N ( G ) (cid:1) . We obtain a C -semigroup of contractions L p (cid:0) V N ( G ) (cid:1) T t −→ L p (cid:0) V N ( G ) (cid:1) which satisfies inequality (5.2). Noncommutative Poisson semigroup.
Let n > be an integer. Recall that F n denotes a freegroup with n generators denoted by g , . . . , g n . A semigroup on L p (cid:0) V N ( F n ) (cid:1) induced by a w ∗ -semigroup of selfadjoint completely positive unital Fourier multipliers on V N ( F n ) satisfies inequality(5.2). An example is provided by the following semigroup. Any g ∈ F n has a unique decompositionof the form g = g k i g k i · · · g k l i l , where l > is an integer, each i j belongs to { , . . . , n } , each k j is a non-zero integer, and i j = i j +1 if j l − . The case when l = 0 corresponds to the unit element g = e F n . By definition, the lengthof g is defined as | g | = | k | + · · · + | k l | . This is the number of factors in the above decomposition of g . For any nonnegative real number t > ,we have a normal unital completely positive selfadjoint map T t : VN( F n ) −→ VN( F n ) λ ( g ) e − t | g | λ ( g ) . These maps define a w*-semigroup ( T t ) t > called the noncommutative Poisson semigroup (see [JMX]for more information). If p < ∞ , this semigroup defines a C -semigroup of contractions L p (cid:0) VN( F n ) (cid:1) T t −→ L p (cid:0) VN( F n ) (cid:1) which satisfies inequality (5.2).24 -Ornstein-Uhlenbeck semigroup. Suppose − q < . Let H be a real Hilbert space and let ( a t ) t > be a C -semigroup of contractions on H . For any t > , let T t = Γ q ( a t ) . Then ( T t ) > isa w*-semigroup of normal unital completely positive maps on the von Neumann algebra Γ q ( H ) . If p < ∞ , this semigroup defines a C -semigroup of contractions L p (cid:0) Γ q ( H ) (cid:1) T t −→ L p (cid:0) Γ q ( H ) (cid:1) (see[JMX] for more information). This semigroup satisfies inequality (5.2).In the case where a t = e − t I H , the semigroup ( T t ) > is the so-called q -Ornstein-Uhlenbeck semi-group. Modular semigroups.
The C -semigroups of isometries satisfy inequality (5.2). Examples areprovided by modular automorphisms semigroups. Here we use noncommutative L p -spaces of a vonNeumann algebra equipped with a distinguished normal faithful state, constructed by Haagerup. Werefer to [PiX], and the references therein, for more information on these spaces. Let M be a vonNeumann algebra with QWEP equipped with a normal faithful state M ϕ −→ C . Let (cid:0) σ ϕt (cid:1) t ∈ R be themodular group of ϕ . If p < ∞ , it is well known that (cid:0) σ ϕt (cid:1) t > induces a C -semigroup of isometries L p ( M ) σ ϕt −−→ L p ( M ) (see [JuX]). This semigroup satisfies inequality (5.2).In the light of Theorem 4.2, it is natural to ask for dilations of unital selfadjoint completely positivesemigroups of Schur multipliers. Actually, these semigroups admit a description which allows us toconstruct a such dilation. Proposition 5.4
Suppose that A is a matrix of M I . For all t > , let T t be the unbounded Schurmultipliers on B (cid:0) ℓ I (cid:1) associated with the matrix h e − ta ij i i,j ∈ I . (5.3) Then the semigroup ( T t ) t > extends to a semigroup of selfadjoint unital completely positive Schurmultipliers B (cid:0) ℓ I (cid:1) T t −→ B (cid:0) ℓ I (cid:1) if and only if there exists a Hilbert space H and a family ( α i ) i ∈ I ofelements of H such that for all t > the Schur multiplier B (cid:0) ℓ I (cid:1) T t −→ B (cid:0) ℓ I (cid:1) is associated with thematrix h e − t k α i − α j k H i i,j ∈ I . In this case, the Hilbert space may be chosen as a real Hilbert space. Moreover, ( T t ) t > is aw*-semigroup.Proof : Now say that each T t is a selfadjoint unital completely positive contraction means that forall t > , the matrix (5.3) defines a real-valued positive definite kernel on I × I in the sense of [BCR,Chapter 3, Definition 1.1] such that for all i ∈ I we have a ii = 0 . Now, the theorem of Schoenberg[BCR, Theorem 2.2] affirms that if ψ is a kernel then e − tψ is a positive definite kernel for all t > ifand only if ψ is a negative definite kernel. Consequently, the last assertion is equivalent to the fact that A defines a real-valued negative definite kernel which vanishes on the diagonal of I × I . Finally, thecharacterization of real-valued definite negative kernel of [BCR, Proposition 3.2] gives the equivalencewith the required description.The assertion concerning the choice of the Hilbert space is clear. Finally, using [JMX, Remark5.2], it is easy to see that ( T t ) t > is a w*-semigroup.The next proposition is inspired by the work [JuX]. Proposition 5.5
Let ( T t ) t > be a w*-semigroup of selfadjoint unital completely positive Schur mul-tipliers on B (cid:0) ℓ I (cid:1) . Then, there exists a hyperfinite von Neumann algebra M equipped with a semifinite ormal faithful trace, a w*-semigroup ( U t ) t > of unital trace preserving *-automorphisms of M , aunital trace preserving one-to-one normal ∗ -homomorphism B (cid:0) ℓ I (cid:1) J −→ M such that T t = E U t J. for any t > , where M E −→ B (cid:0) ℓ I (cid:1) is the canonical faithful normal trace preserving conditional expec-tation associated with J .Proof : By Proposition 5.4, there exists a real Hilbert space H and a family ( α j ) j ∈ I of elements of H such that, for all t > , the Schur multiplier B (cid:0) ℓ I (cid:1) T t −→ B (cid:0) ℓ I (cid:1) is associated with the matrix h e − t k α j − α k k H i j,k ∈ I . Let µ be a gaussian measure on H , i.e. a probability space (Ω , µ ) together with a measurable function Ω w −→ H such that, for all h ∈ H , we have e −k h k H = Z Ω e i h h,w ( ω ) i H dµ ( ω ) where i = − . We define the von Neumann algebra M = L ∞ (Ω) ⊗ B (cid:0) ℓ I (cid:1) . Note that M is ahyperfinite von Neumann algebra. We equip the von Neumann algebra M with the faithful semifinitenormal trace τ M = R Ω · dµ ⊗ Tr . Note that, by [Sak, Theorem 1.22.13], we have a ∗ -isomorphism M = L ∞ (cid:0) Ω , B ( ℓ I ) (cid:1) . We define the canonical injective normal unital ∗ -homomorphism J : B (cid:0) ℓ I (cid:1) −→ L ∞ (Ω) ⊗ B (cid:0) ℓ I (cid:1) x ⊗ x. It is clear that the map J preserves the traces. We denote by M E −→ B (cid:0) ℓ I (cid:1) the canonical faithfulnormal trace preserving conditional expectation of M onto B (cid:0) ℓ I (cid:1) . For all ω ∈ Ω and t > let D t ( ω ) be the diagonal matrix of B (cid:0) ℓ I (cid:1) defined by D t ( ω ) = (cid:20) δ j,k e i √ t h α j ,w ( ω ) i ℓ I (cid:21) j,k ∈ I . Note that, for all t > , the map Ω D t −−→ B ( H ) defines an unitary element of L ∞ (cid:0) Ω , B ( ℓ I ) (cid:1) . Now, forall t > we define the linear map U t : L ∞ (cid:0) Ω , B ( ℓ I ) (cid:1) −→ L ∞ (cid:0) Ω , B ( ℓ I ) (cid:1) f D t f D ∗ t . If t > , it is easy to see that the map U t is a trace preserving ∗ -automorphism of M . For all x ∈ B (cid:0) ℓ I (cid:1) ,we have E U t J ( x ) = E U t (1 ⊗ x )= Z Ω D t ( ω )(1 ⊗ x ) D t ( ω ) ∗ dµ ( ω )= Z Ω (cid:20) e i √ t h α j − α k ,w ( ω ) i H x jk (cid:21) j,k ∈ I dµ ( ω )= h e − t k α j − α k k H x jk i j,k ∈ I = T t ( x ) . t > , we have T t = E U t J. The assertion concerning the regularity of the semigroup is easy and left to the reader.In the same vein, it is not difficult to construct a dilation of the noncommutative Poisson semigroup.The result was already known to F. Lust-Piquard. Moreover, it is easy to dilate the C -semigroups ofcontractions L p (cid:0) Γ q ( H ) (cid:1) Γ q ( a t ) −−−−→ L p (cid:0) Γ q ( H ) (cid:1) , with [SNF, Theorem 8.1].Finally, we have the next result analogue to Corollary 2.6. One can prove this proposition with asimilar argument. Proposition 5.6
Suppose < p < ∞ . Let ( T t ) t > be a C -semigroup of contractions on a non-commutative L p -space L p ( M ) . Suppose that there exists a noncommutative L p -space L p ( N ) where N has QWEP, a C -semigroup ( U t ) t > of isometric operators on L p ( N ) , an isometric embedding L p ( M ) J −→ L p ( N ) and a contractive map L p ( N ) Q −→ L p ( M ) such that, T t = QU t J. for any t > . Then, for all b ∈ L ( R ) with support in R + , we have the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( M ) −→ L p ( M ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) cb,L p ( R ) −→ L p ( R ) . Moreover, if L p ( N ) is a commutative L p -space L p (Ω) , we have, for all b ∈ L ( R ) with support in R + ,the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p (Ω) −→ L p (Ω) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Z + ∞ b ( t ) T t dt (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R ) −→ L p ( R ) . This proposition allows us to give alternate proofs for some results of this section. By example,using [SNF, Theorem 8.1] of dilation of C -semigroups on a Hilbert space, we deduce that the bothinequalities (5.1) and (5.2) are true for p = 2 . By using [Fen], we see that the C -semigroups ofoperators which admit a contractive majorant satisfy inequality (5.1), for < p < ∞ . We begin by observing that the inequalities (1.1) and (1.4) are true for any complex polynomial P of degree 1 and any contraction T . Indeed, suppose that P ( z ) = a + bz , then it is easy to see that k P k = | a | + | b | . Thus, for all p ∞ , we have k P k p = k P k p,S p = | a | + | b | .Now we will determine the real polynomials of higher degree with a similar property. Proposition 6.1
Let P = n X k =0 a k z k be a real polynomial such that a k = 0 for any k n . Thefollowing assertions are equivalent.1. For all < p < ∞ , we have k P k p = n X k =0 | a k | . . For all < p < ∞ , we have k P k p,S p = n X k =0 | a k | .3. There exists < p < ∞ such that k P k p = n X k =0 | a k | .4. There exists < p < ∞ such that k P k p,S p = n X k =0 | a k | .5. The coefficients a k have the same sign or the signs of the a k are alternating (i.e. for any integer k n − we have a k a k +1 ).In this case, for the polynomial P and any contraction T , the inequalities (1.1) and (1.4) are true.Proof : First we will show that k P k = P nk =0 | a k | is equivalent to the last assertion. Recall that k P k = sup | z | =1 | P ( z ) | . On the one hand, for all θ π , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =0 a k e kiθ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = n X k =0 a k cos( kθ ) ! + n X k =0 a k sin( kθ ) ! = n X k =0 a k cos ( kθ ) + 2 X k References [AbA] Y. 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