On bounded operators on a Banach space and derivations on projective tensor algebras (Q2846588)

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 \(\mathcal{U}=\ell^1 \hat{\otimes} \ell^{\infty}\).