Projective Tensor Product Research Articles

Projective tensor products of approximation spaces associated with positive operators

In this paper the projective tensor products of approximation spaces associated with positive operators in Banach spaces are characterized. We show that the tensor products of approximation spaces can be considered as the interpolation spaces generated by $K$-method of real interpolation. The inequalities that provide a sharp estimates of best approximations by analytic vectors of positive operators on projective tensor products are established. Application to spectral approximations of the regular elliptic operators on projective tensor products of Lebesgue spaces is shown.

THE SET OF ELEMENTARY TENSORS IS WEAKLY CLOSED IN PROJECTIVE TENSOR PRODUCTS

Abstract We prove that the set of elementary tensors is weakly closed in the projective tensor product of two Banach spaces. As a result, we answer a question of Rodríguez and Rueda Zoca [‘Weak precompactness in projective tensor products’, Indag. Math. (N.S.)35(1) (2024), 60–75], proving that if $(x_n) \subset X$ and $(y_n) \subset Y$ are two weakly null sequences such that $(x_n \otimes y_n)$ converges weakly in $X \widehat {\otimes }_\pi Y$ , then $(x_n \otimes y_n)$ is also weakly null.

Unconditional bases and Daugavet renormings

We prove that the space ℓ2 – and more generally every infinite dimensional Banach space with an unconditional Schauder basis – can be renormed to have a Daugavet point. As a starting point, we introduce a new diametral notion for points of the unit sphere of Banach spaces, that complements the notion of Δ-points, but is weaker than the notion of Daugavet points. We show that this notion can be used to provide another geometric characterization of the Daugavet property, as well as to recover – and even to provide new – results about Daugavet points in various contexts such as absolute sums of Banach spaces or projective tensor products. Then, we observe that the study of this notion naturally leads to new renorming ideas in the context, and combine these with previous techniques from the literature to show that Daugavet points may exist even in super-reflexive spaces.

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Some properties of 𝑝-limited sets

Karn and Sinha [Glasg. Math. J. 56 (2014), pp. 427–437] introduced the p p -limited ( 1 ≤ p > ∞ 1 \leq p > \infty ) sets (see the definition below). We show that p p -limited sets are preserved by continuous polynomials as well as by the projective tensor product and that scalar-valued polynomials are p p -summable on p p -limited sets. Considering the notion of p p -limited set from the ℓ p \ell _p -valued operators point of view, we introduce in Section \ref{sec:3} two weaker types of p p -limitedness in the setting of Banach lattices and study their basic properties.

Projective freeness and Hermiteness of complex function algebras

The paper studies projective freeness and Hermiteness of algebras of complex-valued continuous functions on topological spaces, Stein algebras, and commutative unital Banach algebras. New sufficient cohomology conditions on the maximal ideal spaces of the algebras are given that guarantee the fulfilment of these properties. The results are illustrated by nontrivial examples. Based on the Borsuk theory of shapes, a new class C of commutative unital complex Banach algebras is introduced (an analog of the class of local rings in commutative algebra) such that the projective tensor product with algebras in C preserves projective freeness and Hermiteness. Some examples of algebras of class C and of other projective free and Hermite function algebras are assembled. These include, e.g., Douglas algebras, finitely generated algebras of symmetric functions, Bohr-Wiener algebras, algebras of holomorphic semi-almost periodic functions, and algebras of bounded holomorphic functions on Riemann surfaces.

Weak precompactness in projective tensor products

We give a sufficient condition for a pair of Banach spaces (X,Y) to have the following property: whenever W1⊆X and W2⊆Y are sets such that {x⊗y:x∈W1,y∈W2} is weakly precompact in the projective tensor product X⊗̂πY, then either W1 or W2 is relatively norm compact. For instance, such a property holds for the pair (ℓp,ℓq) if 1<p,q<∞ satisfy 1/p+1/q≥1. Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of ℓ1 into X⊗̂πY for arbitrary Banach spaces X and Y, in connection with the compactness of all operators from X to Y∗.

Extremal Structure of Projective Tensor Products

We prove that, given two Banach spaces X and Y and bounded, closed convex sets Csubseteq X and Dsubseteq Y, if a nonzero element zin {overline{{{,textrm{co},}}}}(Cotimes D)subseteq Xwidehat{otimes }_pi Y is a preserved extreme point then z=x_0otimes y_0 for some preserved extreme points x_0in C and y_0in D, whenever K(X,Y^*) separates points of X widehat{otimes }_pi Y (in particular, whenever X or Y has the compact approximation property). Moreover, we prove that if x_0in C and y_0in D are weak-strongly exposed points then x_0otimes y_0 is weak-strongly exposed in {overline{{{,textrm{co},}}}}(Cotimes D) whenever x_0otimes y_0 has a neighbourhood system for the weak topology defined by compact operators. Furthermore, we find a Banach space X isomorphic to ell _2 with a weak-strongly exposed point x_0in B_X such that x_0otimes x_0 is not a weak-strongly exposed point of the unit ball of Xwidehat{otimes }_pi X.

Several Remarks on Norm Attainment in Tensor Product Spaces

The aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if Xwidehat{otimes }_pi Y is strongly subdifferentiable and either X or Y has the metric approximation property then every bounded operator from X to Y^* is compact. We also prove that (ell _p(I)widehat{otimes }_pi ell _q(J))^* has the w^*-Kadec-Klee property for every non-empty sets I, J and every 2<p,q<infty , obtaining in particular that the norm of the space ell _p(I)widehat{otimes }_pi ell _q(J) is strongly subdifferentiable. This extends several results of Dantas, Kim, Lee and Mazzitelli. We also find examples of spaces X and Y for which the set of norm-attaining tensors in Xwidehat{otimes }_pi Y is dense but whose complement is dense too.

Some isomorphic properties in projective tensor products

Some isomorphic properties in projective tensor products

On the Fremlin projective tensor product of Banach d-algebras and almost f-algebras

On the Fremlin projective tensor product of Banach d-algebras and almost f-algebras

Narrow operators on tensor products of Köthe spaces

We investigate narrow operators on tensor products of Köthe-Banach spaces. Our first main result asserts that for a bounded linear operator S:F→F on Köthe-Banach space F with an order continuous norm and a linear bounded operator T:L1(μ)→L1(μ), the narrowness at least one of the T or S implies the narrowness for an operator T⊗S:L1(μ)⊗ˆπF→L1(μ)⊗ˆπF defined on the positive projective tensor product of L1(μ) and F. The converse statement holds under an additional assumption of regularity for the operator S:F→F. As a consequence, we resolve an open problem concerning the existence of a Köthe-Banach space on [0,1] with an absolutely continuous norm without an unconditional basis in which the identity operator is a sum of two narrow operators, suggested in [45]. We also supply complementing results for the Köthe-Banach spaces E[X] with mixed norm where E and X are symmetric spaces and fully characterize the spaces E[X] having an unconditional basis.

On weak compactness in projective tensor products

Abstract We study the property of being strongly weakly compactly generated (and some relatives) in projective tensor products of Banach spaces. Our main result is as follows. Let $1 \lt p,q \lt \infty$ be such that $1/p+1/q\geq 1$. Let X (resp., Y) be a Banach space with a countable unconditional finite-dimensional Schauder decomposition having a disjoint lower p-estimate (resp., q-estimate). If X and Y are strongly weakly compactly generated, then so is their projective tensor product $X {\widehat{\otimes}_\pi} Y$.

Topological properties in tensor products of Banach spaces

Given two Banach spaces X and Y, we analyze when the projective tensor product X⊗ˆπY has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a weakly compactly generated (WCG) space or subspace of a Hilbert generated space. For instance, we show that: (i) X⊗ˆπY is WLD if and only if both X and Y are WLD and all operators from X to Y⁎ and from Y to X⁎ have separable range; (ii) X⊗ˆπY is subspace of a WCG space if the same holds for both X and Y under the assumption that every operator from X to Y⁎ is compact; (iii) ℓp(Γ)⊗ˆπℓq(Γ) is subspace of a Hilbert generated space for any 1<p,q<∞ such that 1/p+1/q<1 and for any infinite set Γ. We also pay attention to the injective tensor product X⊗ˆεY. In this case, the stability of property (C) and the property of being WLD turn out to be closely related to the condition that all regular Borel probability measures on the dual ball have countable Maharam type. Along this way, we generalize a result of Plebanek and Sobota that if K is a compact space such that C(K×K) has property (C), then all regular Borel probability measures on K have countable Maharam type. This generalization provides a consistent negative answer to a question of Ruess and Werner about the preservation of the w⁎-angelicity of the dual unit ball under injective tensor products.

On extensions of bilinear maps

Abstract The paper deals with extension of bounded bilinear maps. It gives a necessary and sufficient condition for extending a bounded bilinear map on the Cartesian product of subspaces of Banach spaces. This leads to a full characterization for extension of bounded bilinear maps on the Cartesian product of arbitrary subspaces of Hilbert spaces. Applications concerning projective tensor products are also investigated.

Integrable and absolutely continuous vector-valued functions

The theory of integration for functions with values in a topological vector space is a bit of a messy subject. There are many different choices for the integral, most distinct from one another in general, and with no compelling reason to always adopt one definition or another as the “right” one. No attempt is made here to resolve this question of which integral is the “right” integral, but we overview a few of the common notions and prove a couple of useful properties for the notion of integrability by seminorm. For this notion of integrability, the completeness of the space of integrable functions with values in a complete locally convex space is established. This space is then easily seen to be isomorphic to the completion of the projective tensor product of the usual L1 space of scalar functions with the vector space. Absolute continuity for this notion of integrability is also considered. Here, it is shown that the familiar properties of differentiation for absolutely continuous scalar functions holds in the vector-valued case.

Symmetry and spectral invariance for topologically graded C∗$C^*$‐algebras and partial action systems

A discrete group G ${\sf G}$ is called rigidly symmetric if the projective tensor product between the convolution algebra ℓ 1 ( G ) $\ell ^1({\sf G})$ and any C ∗ $C^*$ -algebra A $\mathcal {A}$ is symmetric. We show that in each topologically graded C ∗ $C^*$ -algebra over a rigidly symmetric group there is a ℓ 1 $\ell ^1$ -type symmetric Banach ∗ $^*$ -algebra, which is inverse closed in the C ∗ $C^*$ -algebra. This includes new general classes, as algebras admitting dual actions and partial crossed products. Results including convolution dominated kernels, inverse closedness with respect with ideals or weighted versions of the ℓ 1 $\ell ^1$ -decay are included. Various concrete examples are presented.

A version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices

Abstract We give a version of Kalton’s theorem for the space of regular homogeneous polynomials on Banach lattices. As applications, we obtain sufficient conditions for the reflexivity of ${\mathcal P}^r(^nE;F)$, the space of regular n-homogeneous polynomials from a Banach lattice E to a Banach lattice F, and sufficient conditions for the positive Grothendieck property of $\hat{\otimes}_{n,s,|\pi|}E$, the n-fold positive projective symmetric tensor product of a Banach lattice E. Moreover, we also prove that these sufficient conditions are also necessary under the bounded regular approximation property.

Convergent star products on cotangent bundles of Lie groups

For a connected real Lie group G we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of G. This star product trivially converges on polynomial functions on T^*G thanks to its homogeneity. We define a nuclear Fréchet algebra of certain analytic functions on T^*G, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter hbar . This nuclear Fréchet algebra is realized as the completed (projective) tensor product of a nuclear Fréchet algebra of entire functions on G with an appropriate nuclear Fréchet algebra of functions on {mathfrak {g}}^*. The passage to the Weyl-ordered star product, i.e. the Gutt star product on T^*G, is shown to preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on hbar .

Daugavet property in projective symmetric tensor products of Banach spaces

We show that all the symmetric projective tensor products of a Banach space X have the Daugavet property provided X has the Daugavet property and either X is an L_1-predual (i.e., X^{*} is isometric to an L_1-space) or X is a vector-valued L_1-space. In the process of proving it, we get a number of results of independent interest. For instance, we characterise “localised” versions of the Daugavet property [i.e., Daugavet points and Delta-points introduced in Abrahamsen et al. (Proc Edinb Math Soc 63:475–496 2020)] for L_1-preduals in terms of the extreme points of the topological dual, a result which allows to characterise a polyhedrality property of real L_1-preduals in terms of the absence of Delta-points and also to provide new examples of L_1-preduals having the convex diametral local diameter two property. These results are also applied to nicely embedded Banach spaces [in the sense of Werner (J Funct Anal 143:117–128, 1997)] so, in particular, to function algebras. Next, we show that the Daugavet property and the polynomial Daugavet property are equivalent for L_1-preduals and for spaces of Lipschitz functions. Finally, an improvement of recent results in Rueda Zoca (J Inst Math Jussieu 20(4):1409–1428, 2021) about the Daugavet property for projective tensor products is also obtained.