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

Frank, Michael

Math. Scand. 80, No. 2, 313-319 (1997).

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

Keywords:

isometries; *-isomorphisms; Banach \(C^*\)-module; Hilbert \(C^*\)-modules; multiplier \(C^*\)-algebras