Isomorphisms of Hilbert C*-Modules and $\ast$-Isomorphisms of Related Operator C*-Algebras

Let $\mathcal{M}$ be a Banach C*-module over a C*-algebra $A$ carrying two $A$-valued inner products $<\mathrm{.},\mathrm{.}{>}_1$, $<\mathrm{.},\mathrm{.}{>}_2$ which induce equivalent to the given one norms on $\mathcal{M}$. Then the appropriate unital C*-algebras of adjointable bounded $A$-linear operators on the Hilbert $A$-modules ${ {\cal M}, < .,. >_1 }$and${ {\cal M}, < .,. >_2 }$areshowntobe$$-isomorphicifandonlyifthereexistsabounded$A$-linearisomorphism$S$ofthesetwoHilbert$A$-modulessatisfyingtheidentity$< .,. >_2 \equiv < S(.),S(.) >_1$. 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 $\ast$-isomorphic C*-algebras of ‘‘compact’’/adjointable bounded module operators is indicated.