scholarly journals Properties (V) and (wV) in Projective Tensor Products

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Ioana Ghenciu
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Sufficient Conditions ◽  
Tensor Products ◽  
Weakly Compact ◽  
Projective Tensor Product ◽  
Projective Tensor

We give sufficient conditions for a subset of K(X,Y*)=L(X,Y*) to be relatively weakly compact. A Banach space X has property (V) (resp., (wV)) if every V-subset of X* is relatively weakly compact (resp., weakly precompact). We prove that the projective tensor product X⊗πY has property (V) (resp., (wV)), when X has property (V) (resp., (wV)), Y has property (V), and W(X,Y*)=K(X,Y*).

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

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract 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 *.

scholarly journals Polynomial Grothendieck properties

1995 ◽  
Vol 37 (2) ◽  
pp. 211-219 ◽  
Author(s):  
Manuel González ◽  
Joaquí M. Gutiérrez
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Homogeneous Polynomial ◽  
Bounded Operator ◽  
Weakly Compact ◽  
Completely Continuous ◽  
Projective Tensor Product ◽  
Linear Bounded Operator ◽  
Grothendieck Property ◽  
Projective Tensor

AbstractA Banach space sE has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space (kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric A>fold projective tensor product of £(i.e., the predual of (kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product EF is also characterized. Moreover, the Grothendieck property of E is described in terms of sequences of polynomials. Finally, it is shown that if every operator from E into c0 is completely continuous, then so is every polynomial between these spaces.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract 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.

scholarly journals α-Derivations and their norm in projective tensor products ofΓ-Banach algebras

1998 ◽  
Vol 21 (2) ◽  
pp. 359-368 ◽  
Author(s):  
T. K. Dutta ◽  
H. K. Nath ◽  
R. C. Kalita
Keyword(s):  
Tensor Product ◽  
Arbitrary Element ◽  
Banach Algebras ◽  
Tensor Products ◽  
Projective Tensor Product ◽  
Projective Tensor

Let(V,Γ)and(V′,Γ′)be Gamma-Banach algebras over the fieldsF1andF2isomorphic to a fieldFwhich possesses a real valued valuation, and(V,Γ)⊗p(V′,Γ′), their projective tensor product. It is shown that ifD1andD2areα- derivation andα′- derivation on(V,Γ)and(V′,Γ′)respectively andu=∑1x1⊗y1, is an arbitrary element of(V,Γ)⊗p(V′,Γ′), then there exists anα⊗α′- derivationDon(V,Γ)⊗p(V′,Γ′)satisfying the relationD(u)=∑1[(D1x1)⊗y1+x1⊗(D2y1)]and possessing many enlightening properties. The converse is also true under a certain restriction. Furthermore, the validity of the results‖D‖=‖D1‖+‖D2‖andsp(D)=sp(D1)+sp(D2)are fruitfully investigated.

scholarly journals Some properties of the projective tensor product UX derived from those of U and X

2006 ◽  
Vol 73 (1) ◽  
pp. 37-45 ◽  
Author(s):  
Patrick N. Dowling
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Unconditional Basis ◽  
Complex Banach Space ◽  
Continuity Property ◽  
Complete Continuity ◽  
Projective Tensor Product ◽  
Projective Tensor

Let X be a real or complex Banach space and let U be a Banach space with an unconditional basis. We show that the projective tensor product of U and X, UX, has the complete continuity property (respectively, the analytic complete continuity property) whenever U and X have the complete continuity property (respectively, the analytic complete continuity property). More general versions of these results are also obtained. Moreover, the techniques applied here to the projective tensor product, can also be used to establish some Banach space properties of the Fremlin projective tensor product.

A New Approach to Operator Spaces

1991 ◽  
Vol 34 (3) ◽  
pp. 329-337 ◽  
Author(s):  
Edward G. Effros ◽  
Zhong-Jin Ruan
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Operator Space ◽  
Operator Spaces ◽  
New Approach ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Space Concepts ◽  
Injective Operator

AbstractThe authors previously observed that the space of completely bounded maps between two operator spaces can be realized as an operator space. In particular, with the appropriate matricial norms the dual of an operator space V is completely isometric to a linear space of operators. This approach to duality enables one to formulate new analogues of Banach space concepts and results. In particular, there is an operator space version ⊗μ of the Banach space projective tensor product , which satisfies the expected functorial properties. As is the case for Banach spaces, given an operator space V, the functor W |—> V ⊗μ W preserves inclusions if and only if is an injective operator space.

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