  For unital $C^*$-algebras $A$ and $B$, we completely characterize the
isometric ($*$-) automorphisms of their Banach space projective tensor product
$A\otimes^\gamma B$. This leads to the characterization of inner and outer
isometric $*$-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 $C^*$-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
$C^*$-algebra, and $A$ is any $C^*$-algebra.
