Journal of Operator Theory

Volume 59, Issue 2, Spring 2008 pp. 431-434.

Hilbert

$C^*$ -modules and

$*$ -isomorphisms

Authors: Mohammad B. Asadi
Author institution: Department of Mathematical Sciences, Shahed University, Tehran, Iran

Summary: In this study, it is shown that if

$E_1$ and

$E_2$ are Hilbert

$C^*$ -modules over a

$C^*$ -algebra of (not necessarily all) compact operators and

$\Phi$ is a

$*$ -isomorphism between

$C^*$ -algebras

$\mathcal{L}(E_1)$ and

$\mathcal{L}(E_2)$ , then

$\Phi$ is in the form

$\mathrm{Ad} U$ , for some unitary operator

$U: E_1 \to E_2$ , and so

$E_1$ and

$E_2$ are isomorphic as Hilbert

$C^*$ -modules. This implies that if

$C^*$ -algebras

$\A$ and

$K(H)$ are strongly Morita equivalent then the Picard group of

$\A$ is trivial.

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