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# Journal of Operator Theory

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

Hilbert $C^*$-modules and $*$-isomorphisms

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.