A Characterization of a Local Vector Valued Bollobás Theorem

In this paper, we are interested in giving two characterizations for the so-called property L$$_{o,o}$$ o , o , a local vector valued Bollobás type theorem. We say that (X, Y) has this property whenever given $$\varepsilon > 0$$ ε > 0 and an operador $$T: X \rightarrow Y$$ T : X → Y , there is $$\eta = \eta (\varepsilon , T)$$ η = η ( ε , T ) such that if x satisfies $$\Vert T(x)\Vert > 1 - \eta $$ ‖ T ( x ) ‖ > 1 - η , then there exists $$x_0 \in S_X$$ x 0 ∈ S X such that $$x_0 \approx x$$ x 0 ≈ x and T itself attains its norm at $$x_0$$ x 0 . This can be seen as a strong (although local) Bollobás theorem for operators. We prove that the pair (X, Y) has the L$$_{o,o}$$ o , o for compact operators if and only if so does $$(X, \mathbb {K})$$ ( X , K ) for linear functionals. This generalizes at once some results due to D. Sain and J. Talponen. Moreover, we present a complete characterization for when $$(X \widehat{\otimes }_\pi Y, \mathbb {K})$$ ( X ⊗ ^ π Y , K ) satisfies the L$$_{o,o}$$ o , o for linear functionals under strict convexity or Kadec–Klee property assumptions in one of the spaces. As a consequence, we generalize some results in the literature related to the strongly subdifferentiability of the projective tensor product and show that $$(L_p(\mu ) \times L_q(\nu ); \mathbb {K})$$ ( L p ( μ ) × L q ( ν ) ; K ) cannot satisfy the L$$_{o,o}$$ o , o for bilinear forms.

SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

ARENS REGULARITY OF PROJECTIVE TENSOR PRODUCT

In this paper, we study approximate identity properties, some propositions from Baker, Dales, Lau in general situations and we establish some relationships between the topological centers of module actions and factorization properties with some results in group algebras. We consider under which sufficient and necessary conditions the Banach algebra $A\widehat{\otimes}B$ is Arens regular.

The dual of the space of bounded operators on a Banach space

Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.