On Blecher's characterization of Hilbert C^*-modules
by
Franka Miriam Brückler
Department of Mathematics, University of Zagreb

A Hilbert C^*-module is a generalized Hilbert space: the inner product is allowed to take values in an arbitrary C*-algebra. D.P. Blecher has shown (1995) that Hilbert C^*-modules are characterized as Banach modules V over a (nondegenerate) C^*-algebra A subset or equal B(H) for which the construction V\otimes_hAH^c yields a Hilbert space H_V such that V can be embedded completely isometrically in B(H, H_V).

For Hilbert C^*-modules V over the C^*-algebra K(H) of all compact operators on a Hilbert space H, D. Baki\'c and B. Guljas have shown (1999) that 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. The space V_e has further useful properties, e.g. it enables one to find an orthonormal basis for such a module V (orthonormal bases don't exist for general Hilbert C^*-modules). It turns out that this Hilbert space V_e is isometrically isomorphic to the Hilbert space H_V from the Blecher theorem.

Further, D. Baki\' c and B. Guljas defined a generalization of the notion of the multiplier algebra of a C^*-algebra: the V-strict completion of a Hilbert C^*-module V. This strict completion can be realized in the algebra B(H, H_V), the same one in which V can be embedded by Blecher's theorem.

Date received: April 1, 2001

Copyright © 2001 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 # cagt-10.