On Banach space projective tensor product of $C^*$-algebras

Abstract

We analyze certain algebraic structures of the Banach space projective tensor product of $C^*$-algebras which are comparable with their known counterparts or the Haagerup tensor product and the operator space projective tensor product of $C^*$-algebras. Highlights of this analysis include (a) injectivity of the Banach space projective tensor product when restricted to the tensor products of $C^*$-algebras, (b) detailed structure of closed ideals of $A \otimes_{\gamma} B$ in terms of those of $A$ and $B$, (c) identification of certain spaces of ideals of $A \otimes_{\gamma} B$ in terms of those of $A$ and $B$ from the perspective of hull-kernel topology, and (d) identification of the center of $A \otimes_{\gamma} B$ with $Z(A) \otimes_{\gamma} Z(B)$, where $A$ and $B$ are $C^*$-algebras.