On a characterisation of Hilbert C^*-modules
by
Franka Miriam Brückler
Univ. of Zagreb, Croatia

A Hilbert C^*-module is a generalised Hilbert space: the inner product is allowed to take values in an arbitrary C^*-algebra. For Hilbert spaces there is a metric characterisation, the parallelogram identity, which allows the inner product to be reconstructed from the norm. For Hilbert C^*-modules the situation is more complicated, but there is also a metric characterisation: D.P. Blecher has shown (1995) that a Hilbert C^*-module V over a C^*-algebra A subset or equal B(H) is characterised by a Hilbert space H_V, obtained as a tensor product, and a map \phi allowing the Hilbert C^*-module to be embedded into bounded operators between the Hilbert spaces H and H_V, such that \phi(x)^*\phi(x) in A for all x in V. In this case the inner product can be reconstructed via \phi: < x | y > =\phi(x)^*\phi(y) for x, y in V.

Many of the properties of Hilbert spaces cannot be obtained for general Hilbert C^*-modules, e.g. the decomposition into orthogonal complements. A special class of Hilbert C^*-modules, for which many of the difficulties can be resolved, are Hilbert C^*-modules over C^*-algebras of compact operators on a Hilbert space. One of the nice properties of such modules, as D. Baki\'c and B. Guljas have shown in 1999, is the following: if V is a Hilbert C^*-module over the C^*-algebra K(H) of all compact operators on a Hilbert space H, then there is a Hilbert subspace V_e of V such that the C^*-algebra of all adjointable (with respect to the Hilbert C^*-module inner product) maps on V is isomorphic to the C^*-algebra of all bounded linear maps on V_e, via the restriction map. It turns out that this Hilbert space V_e is isometrically isomorphic to the Hilbert space from the Blecher theorem.

Date received: May 2, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-68.