Projective Tensor Products of $C^*$-algebras

For $C^*$-algebras $A$ and $B$, we study the bi-continuity of the canonical embedding of $A^{**}\ot_γ B^{**}$ ($A^{**}\hat{\ot} B^{**}$) into $(A \ot_γ B)^{**}$ (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.

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