## Isomorphisms of Hilbert $$C^*$$-modules and $$*$$-isomorphisms of related operator $$C^*$$-algebras.(English)Zbl 0892.46057

Let $${\mathcal M}$$ be a Banach $$C^*$$-module over a $$C^*$$-algebra $$A$$ carrying two $$A$$-valued inner products $$\langle.,. \rangle_1$$, $$\langle., \rangle_2$$ which induce norms on $${\mathcal M}$$ equivalent to the given one. Then the appropriate unital $$C^*$$-algebras of adjoinable bounded $$A$$-linear operators on $$\{{\mathcal M}, \langle.,. \rangle_1 \}$$, $$\{{\mathcal M}, \langle .,. \rangle_2 \}$$ are shown to be $$*$$-isomorphic if and only if there exists a bounded $$A$$-linear isomorphism $$S$$ of these two Hilbert $$C^*$$-modules satisfying the identity $$\langle .,. \rangle_1 \equiv \langle S(.), S(.) \rangle_2$$. 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 coefficients and of “compact”/adjointable operators is indicated. The result is closely related to the problem whether a multiplier $$C^*$$-algebra allows to recover the original $$C^*$$-algebra without identity in a unique way, or not.
Reviewer: M.Frank (Leipzig)

### MSC:

 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46L05 General theory of $$C^*$$-algebras
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