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

### New Zealand Journal of Mathematics

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

CONICET- UNCPBA,

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 If the underlying space has no such basis our results are no longer applicable.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.