  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.
