Projective Tensor Products of $C^*$-algebras

Abstract

For $C^*$-algebras $A$ and $B$, we study the bi-continuity of the canonical embedding of $A^{**}\ot_{\gamma} B^{**}$ ($A^{**}\hat{\ot} B^{**}$) into $(A \ot_{\gamma} B)^{**}$ (resp. $(A \hat{\ot} B)^{**}$), and its isomorphism. Ideal structure of $A\hat{\ot} B$ has been obtained in case $A$ or $B$ has only finitely many closed ideals.