On bounded operators on a Banach space and derivations on projective tensor algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1294.47054
author: Ana Paula Madrid
publication date: 2013-09-06T00:00:00Z
full work available at URL: On Bounded Operators on a Banach Space and Derivations on Projective Tensor Algebras
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47B48
zbMATH DE Number: 6206257
zbMATH Keywords: shrinking basis of a Banach space
zbMATH Keywords: associated sequence of coefficient functionals
zbMATH Keywords: projective tensor product of Banach spaces
MaRDI profile type: Publication
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title: On bounded operators on a Banach space and derivations on projective tensor algebras
published in: New Zealand Journal of Mathematics
review text: Let \(X\) be a complex Banach space. The equation NEWLINE\[NEWLINE \left( x_1 \otimes x_{1}^{\ast} \right) \left( x_2 \otimes x_{2}^{\ast} \right)\triangleq\langle x_2,x_{1}^{\ast} \rangle x_{1} \otimes x_{2}^{\ast} NEWLINE\]NEWLINE defines a product that turns the projective tensor product \(\mathcal{U}=X\hat{\otimes} X^{\ast}\) into a Banach algebra. Consider the continuous map \(D : B(X) \rightarrow B(\mathcal{U})\) defined by \(D(T) = T\otimes I_{X^{\ast}} - I_{X}\otimes T^{\ast}\). The main result of this paper is a description of those inner derivations on \(\mathcal{U}\) that are of the form \(D(T)\) for some \(T\in B(X)\). This description is given only for Banach spaces with a shrinking basis. For the Banach space \(X=\ell^1\), the author is able to give a description of the derivations of the corresponding algebra \(\mathcal{U}=\ell^1 \hat{\otimes} \ell^{\infty}\).
reviewed by: Jerónimo Alaminos Prats
description: scientific article; zbMATH DE number 6206257