Isomorphisms of Hilbert $C^*$-modules and *-isomorphisms of related operator $C^*$-algebras.

Let $\cal M$ be a Banach C*-module over a C*-algebra $A$ carrying two $A$-valued inner products $ _1$, $ _2$ which induce equivalent to the given one norms on $\cal M$. Then the appropriate unital C*-algebras of adjointable bounded $A$-linear operators on the Hilbert $A$-modules $\{ {\cal M}, _1 \}$ and $\{ {\cal M}, _2 \}$ are shown to be $*$-isomorphic if and only if there exists a bounded $A$-linear isomorphism $S$ of these two Hilbert $A$-modules satisfying the identity $ _2 \equiv _1$. This result extends other equivalent descriptions due to L.~G.~Brown, H.~Lin and E.~C.~Lance. An example of two non-isomorphic Hilbert C*-modules with $*$-isomorphic C*-algebras of ''compact''/adjointable bounded module operators is indicated.