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

Abstract

Let $\mathcal{M}$ 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 $\mathcal{M}$. Then the appropriate unital C*-algebras of adjointable bounded $A$-linear operators on the Hilbert $A$-modules $\{\mathcal{M},<.,.{>}_1\}$ and $\{\mathcal{M},<.,.{>}_2\}$ are shown to be $\ast$-isomorphic if and only if there exists a bounded $A$-linear isomorphism $S$ of these two Hilbert $A$-modules satisfying the identity $<.,.{>}_2\equiv <S(.),S(.){>}_1$. 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.

5 pages, LaTeX file

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