Injective Tensor Product Research Articles

Geometry of the Banach spaces C(βN ×K,X) for compact metric spaces K

In this paper we provide the complete isomorphic classication of the spaces C(�N × K,lp) of all continuous lp-valued functions, 1 � p < 1, dened on the topological product of the Stone-Cech compactication of the natural numbers N and an arbitrary innite compact metric space K. In order to do this, werst prove that c0 is the only innite dimensional separable C(K) space, Z, up to an isomorphism, which satises each one of the following statements: • Z is a quotient of C(�N,lp) for every 1 < p < 1. • Z is isomorphic to a complemented subspace of C(�N,l1). • C(�N,lp) is isomorphic to the injective tensor product of itself and Z, for every 1 � p < 1.

Interpolation of vector measures

Let (Ω, Σ) be a measurable space and m0: Σ → X0 and m1: Σ → X1 be positive vector measures with values in the Banach Kothe function spaces X0 and X1. If 0 < α < 1, we define a new vector measure [m0, m1]α with values in the Calderon lattice interpolation space X01−gaX1α and we analyze the space of integrable functions with respect to measure [m0, m1]α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of Lp-spaces. Since each p-convex order continuous Kothe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.

VECTOR MEASURES OF BOUNDED SEMIVARIATION AND ASSOCIATED CONVOLUTION OPERATORS

AbstractLet G be a compact metrizable abelian group, and let X be a Banach space. We characterize convolution operators associated with a regular Borel X-valued measure of bounded semivariation that are compact (resp; weakly compact) from L1(G), the space of integrable functions on G into L1(G) X, the injective tensor product of L1(G) and X. Along the way we prove a Fourier Convergence theorem for vector measures of relatively compact range that are absolutely continuous with respect to the Haar measure.

The Grothendieck property for injective tensor products of Banach spaces

Let X be a Banach space with the Grothendieck property, Y a reflexive Banach space, and let X ⊗ɛY be the injective tensor product of X and Y. (a) If either X** or Y has the approximation property and each continuous linear operator from X* to Y is compact, then X ⊗ɛY has the Grothendieck property. (b) In addition, if Y has an unconditional finite dimensional decomposition, then X ⊗ɛY has the Grothendieck property if and only if each continuous linear operator from X* to Y is compact.

On p-approximation properties for p-operator spaces

This paper has a two-fold purpose. Let 1 < p < ∞ . We first introduce the p-operator space injective tensor product and study various properties related to this tensor product, including the p-operator space approximation property, for p-operator spaces on L p -spaces. We then apply these properties to the study of the pseudofunction algebra PF p ( G ) , the pseudomeasure algebra PM p ( G ) , and the Figà–Talamanca–Herz algebra A p ( G ) of a locally compact group G. We show that if G is a discrete group, then most of approximation properties for the reduced group C ∗-algebra C λ ∗ ( G ) , the group von Neumann algebra VN ( G ) , and the Fourier algebra A ( G ) (related to amenability, weak amenability, and approximation property of G) have the natural p-analogues for PF p ( G ) , PM p ( G ) , and A p ( G ) , respectively. The p-completely bounded multiplier algebra M cb A p ( G ) plays an important role in this work.

Weakly open sets in the unit ball of some Banach spaces and the centralizer

We show that every Banach space X whose centralizer is infinite-dimensional satisfies that every non-empty weakly open set in B Y has diameter 2, where Y = ⊗ ˆ N , s , π X ( N-fold symmetric projective tensor product of X, endowed with the symmetric projective norm), for every natural number N. We provide examples where the above conclusion holds that includes some spaces of operators and infinite-dimensional C ∗ -algebras. We also prove that every non-empty weak ∗ open set in the unit ball of the space of N-homogeneous and integral polynomials on X has diameter two, for every natural number N, whenever the Cunningham algebra of X is infinite-dimensional. Here we consider the space of N-homogeneous integral polynomials as the dual of the space ⊗ ˆ N , s , ε X ( N-fold symmetric injective tensor product of X, endowed with the symmetric injective norm). For instance, every infinite-dimensional L 1 ( μ ) satisfies that its Cunningham algebra is infinite-dimensional. We obtain the same result for every non-reflexive L-embedded space, and so for every predual of an infinite-dimensional von Neumann algebra.

On a formula of Le Merdy for the complex interpolation of tensor products

C. Le Merdy in (Proc. Amer. Math. Soc. 126 (1998), 715- 719) proved the following complex interpolation formula for injective tensor products: ('2 ˜ '1,'2 ˜ '1) 1 = S4. We investigate whether related formulas hold when considering arbitrary 0 <

Reflexivity and the Grothendieck property for positive tensor products of Banach lattices-II

Let ϕ be an Orlicz function such that ϕ and its complementary function ϕ* satisfy the Δ2-condition, let ℓϕ be an Orlicz sequence space associated to ϕ, and let X be a Banach lattice. Then ℓϕ ⊗F X (respectively, ℓϕ ∼⊗i X), the Fremlin projective (respectively, the Wittstock injective) tensor product of ℓϕ and X, has reflexivity or the Grothendieck property if and only if X has the same property and each positive linear operator from ℓϕ (respectively, from ℓϕ*) to X* (respectively, to X**) is compact. *The author is partially supported by the NSF of China, Grant No. 10871213. † The author is supported by the NSF of China, Grant No. 10671048.

Reflexivity and the Grothendieck property for positive tensor products of Banach lattices-I

Let X be a Banach lattice and p, p′ be real numbers such that 1 < p, p′<∞ and 1/p + 1/p′ = 1. Then $${\ell_p\hat{\otimes}_FX}$$ (respectively, $${\ell_p\tilde{\otimes}_{i}X}$$ ), the Fremlin projective (respectively, the Wittstock injective) tensor product of l p and X, has reflexivity or the Grothendieck property if and only if X has the same property and each positive linear operator from l p (respectively, from l p′) to X* (respectively, to X**) is compact.

The Schur property on projective and injective tensor products

The problem of whether the Schur property is passed from a Banach space to its (symmetric) projective n-fold tensor product is reformu lated in the language of polynomial ideals. As a result, a very closely related question is solved in the negative. It is also proved that the injective tensor product of infrabarrelled locally convex spaces with the Schur property has the Schur property as well.

Projections on tensor products of Banach spaces

We characterize norm hermitian operators on classes of tensor products of Banach spaces and derive results for the particular settings of injective and projective tensor products. We provide a characterization of bi-circular and generalized bi-circular projections on tensor products of Banach spaces supporting only dyadic surjective isometries.

Banach spaces with the Daugavet property, and the centralizer

We introduce representable Banach spaces, and prove that the class R of such spaces satisfies the following properties: (1) Every member of R has the Daugavet property. (2) It Y is a member of R , then, for every Banach space X, both the space L ( X , Y ) (of all bounded linear operators from X to Y) and the complete injective tensor product X ⊗ ˆ ϵ Y lie in R . (3) If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, and for most vector space topologies τ on Y, the space C ( K , ( Y , τ ) ) (of all Y-valued τ-continuous functions on K) is a member of R . (4) If K is a perfect compact Hausdorff topological space, then, for every Banach space Y, most C ( K , Y ) -superspaces (in the sense of [V. Kadets, N. Kalton, D. Werner, Remarks on rich subspaces of Banach spaces, Studia Math. 159 (2003) 195–206]) are members of R . (5) All dual Banach spaces without minimal M-summands are members of R .

Complexification of the projective and injective tensor products

We show that the Taylor (resp. Bochnak) complexification of the injective (projective) tensor product of any two real Banach spaces is isometrically isomorphic to the injective (projective) tensor product of the Taylor (Bochnak) complexifications of the t

Notes on the geometry of the space of polynomials

AbstractWe show that the symmetric injective tensor product space is not complex strictly convex if E is a complex Banach space of dim E ≥ 2 and if n ≥ 2 holds. It is also reproved that ℓ∞ is finitely represented in if E is infinite‐dimensional and if n ≥ 2 holds, which was proved in the other way in [3]. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

One-sided interpolation of injective tensor products of Banach spaces

The Kouba formula for the complex interpolation of injective tensor products requires all spaces involved to be at least of cotype $2$. We show that this can be weakened when one side of the tensor products is fixed.

An operator space approach to condition C

We show that there exists a natural embedding from the tensor product V ∗ ∗ ⊗ W ∗ ∗ of the biduals of operator spaces V and W into the bidual ( V ⊗ ˇ W ) ∗ ∗ of the injective tensor product of V and W, which is separately weak ∗ continuous. From this, we define condition C for operator spaces.

Complemented copies of ℓp spaces in tensor products

We give sufficient conditions on Banach spaces X and Y so that their projective tensor product X ⊗π Y, their injective tensor product X ⊗ɛ Y, or the dual (X ⊗π Y)* contain complemented copies of ℓp.

On some properties of the space of tensor integrable functions

The tensor integral of a vector valued function f: Ω → X with respect to a countably additive vector valued measure v: Σ → Y has been defined by Stefansson in [14] and he has investigated many of its properties. The integral is an element of the injective tensor product X\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ \otimes } \)Y. We study the Banach space L1(v, X, Y) of all \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ \otimes } \)-integrable functions and discuss many properties of this space. We also study the space w-L1(v, X, Y) of all weakly \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ \otimes } \)-integrable functions.

Some Properties of the Injective Tensor Product of Banach Spaces

Let X and Y be Banach spaces such that X has an unconditional basis. Then X $$ \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{ \otimes } $$ Y , the injective tensor product of X and Y , has the Radon–Nikodym property (respectively, the analytic Radon–Nikodym property, the near Radon–Nikodym property, non-containment of a copy of c 0, weakly sequential completeness) if and only if both X and Y have the same property and each continuous linear operator from the predual of X to Y is compact.

A generalization of a theorem of A. Grothendieck

In this article we characterize the quasi-barrelledness of the projective tensor product of a coechelon space of type one k 1(A) with a Fréchet space, including homological conditions as exactness properties of the corresponding tensor product functor k 1(A) ·: ℱ → ℒ︁, acting from the category of Fréchet spaces to the category of linear spaces, resp. the vanishing of its first right derivative Tor1π (k 1(A),.). This generalizes and extends a classical theorem of A. Grothendieck ([13, Chap. II, §4, No. 3, Theorem 15]). Further we present an analogous theorem for complete coechelon spaces of type zero and the injective tensor product and results concerning the stronger property of barrelledness. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)