On Bounded Operators on a Banach Space and Derivations on Projective Tensor Algebras

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New Zealand Journal of Mathematics

Vol. 43, (2013), Pages 23-29


Ana Paula Madrid

CONICET- UNCPBA,

Facultad de Ciencias Exactas,

NUCOMPA.






Abstract We consider a Banach space \mathcal{X} endowed with a shrinking basis. Then we describe the structure of derivations on the projective Banach algebra \mathcal{X}\hat{\mathfrak{\otimes}}\mathcal{X}^{*} which are induced by bounded linear operators on \mathcal{X}. If the underlying space has no such basis our results are no longer applicable. However, if \mathcal{X} is the space of absolutely convergent complex series we establish a relationship between bounded derivations on l^{1}\left( \mathcal{X}^{*}\right) and bounded derivations on \mathcal{X} \hat{\mathfrak{\otimes}}\mathcal{X}^{*}.

Keywords Shrinking basis of a Banach space. Associated sequence of coefficient functionals. Projective tensor product of Banach spaces.

Classification (MSC2000) 47B47, 47B48.

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