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

TL;DR

This paper characterizes when two Hilbert C*-modules have $*$-isomorphic algebras of adjointable operators, linking module isomorphisms to algebra isomorphisms, extending previous results in the theory of operator modules.

Contribution

It establishes a necessary and sufficient condition for $*$-isomorphism of operator algebras based on the existence of a bounded module isomorphism satisfying a specific inner product relation.

Findings

$*$-isomorphic operator algebras correspond to module isomorphisms with compatible inner products

Extension of previous results by Brown, Lin, and Lance

Example of non-isomorphic modules with $*$-isomorphic operator algebras

Abstract

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 $< .,. >_2 \equiv < S(.),S(.) >_1$. 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 $*$-isomorphic C*-algebras of ''compact''/adjointable bounded module operators is indicated.