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

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

1995-05-09

arxiv.org/abs/funct-an/9505001v1

Math. Scand. , 80(1997), 313-319

Mathematics

Operator Algebras

5 pages, LaTeX file

Scientific paper

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.

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