Automorphisms of Banach space projective tensor product of C*-algebras

Abstract

For unital $C^{\ast }$-algebras $A$ and $B$, we completely characterize the isometric ($\ast$-) automorphisms of their Banach space projective tensor product $A{\otimes }^{\gamma }B$. This leads to the characterization of inner and outer isometric $\ast$-automorphisms of $A{\otimes }^{\gamma }B$, as well. As an application, we provide a partial affirmative answer to a question posed by Kaijser and Sinclair, viz., we prove that for unital $C^{\ast }$-algebras $A$ and $B$, the set of norm-one unitaries of $A{\otimes }^{\gamma }B$ coincides with $U(A)\otimes U(B)$, where $U(A)$ is the unitary group of $A$. We also establish the fact that the relative commutant of $A{\otimes }^{\gamma }\mathbb{C}1$ in $A{\otimes }^{\gamma }B$ is same as $Z(A){\otimes }^{\gamma }B$, where $B$ is a subhomogenous unital $C^{\ast }$-algebra, and $A$ is any $C^{\ast }$-algebra.

8 pages