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 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 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.

Johnson pseudo-contractibility of second duals of certain Banach algebras

2019 ◽  
pp. 2150012
Author(s):  
A. Sahami ◽  
E. Ghaderi ◽  
S. M. Kazemi Torbaghan ◽  
B. Olfatian Gillan
Keyword(s):  
Tensor Product ◽  
Matrix Algebra ◽  
Banach Algebras ◽  
Amenable Group ◽  
Semigroup Algebra ◽  
Archimedean Semigroup ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
The Matrix ◽  
Second Dual

In this paper, we study Johnson pseudo-contractibility of second dual of some Banach algebras. We show that the semigroup algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is a finite amenable group, where [Formula: see text] is an archimedean semigroup. We also show that the matrix algebra [Formula: see text] is Johnson pseudo-contractible if and only if [Formula: see text] is finite. We study Johnson pseudo-contractibility of certain projective tensor product second duals Banach algebras.

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 Compact Hermitian operators on projective tensor products of Banach algebras

2002 ◽  
Vol 29 (3) ◽  
pp. 167-178
Author(s):  
T. K. Dutta ◽  
H. K. Nath ◽  
H. K. Sarmah
Keyword(s):  
Tensor Product ◽  
Analytic Semigroup ◽  
Approximation Property ◽  
Banach Algebras ◽  
Hermitian Operator ◽  
Open Problems ◽  
Projective Tensor Product ◽  
Finite Dimensional ◽  
Projective Tensor ◽  
Compact Analytic Semigroup

LetUandVbe, respectively, an infinite- and a finite-dimensional complex Banach algebras, and letU⊗pVbe their projective tensor product. We prove that (i) every compact Hermitian operatorT1onUgives rise to a compact Hermitian operatorTonU⊗pVhaving the properties that‖T1‖=‖T‖andsp(T1)=sp(T); (ii) ifUandVare separable andUhas Hermitian approximation property(HAP), thenU⊗pVis also separable and hasHAP; (iii) every compact analytic semigroup(CAS)onUinduces the existence of aCASonU⊗pVhaving some nice properties. In addition, the converse of the above results are discussed and some open problems are posed.

scholarly journals Types of Radon-Nikodym properties for the projective tensor product of Banach spaces

2003 ◽  
Vol 47 (4) ◽  
pp. 1303-1326 ◽  
Author(s):  
Qingying Bu ◽  
Joe Diestel ◽  
Patrick Dowling ◽  
Eve Oja
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor
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