Extensions of Hilbert C^*-modules
by
Damir Bakić
Department of Mathematics, University of Zagreb
Coauthors: Boris Guljaš

A closed submodule X of a Hilbert C^*-module V over a C^*-algebra A is said to be an ideal submodule of V if X is of the form X=VI for some ideal I in A. The quotient V/X over an ideal submodule X possesses a natural Hilbert A/I-module structure.

An (essential) extension of a Hilbert A-module V is a quadruple (W, B, \Phi, j) consisting of a Hilbert B-module W, an injective morphism of C^*-algebras j: A --> B such that Im j is an (essential) ideal in B, and an isometric morphism of Hilbert C^*-modules \Phi: V --> W such that Im \Phi is the ideal submodule of W associated with Im j. Each extension (W, B, \Phi, j) of V induces the exact sequence 0 --> V --> W --> W/Im \Phi --> 0 of Hilbert C^*-modules.

One shows that for each Hilbert A-module V there exists the largest essential extension (V_d, B(A), \Gamma, \gamma) such that for any other essential extension (W, B, \Phi, j, ) of V one can embed W into V_d. It will be shown that the C^*-algebras of all adjointable operators acting on V and V_d, respectively, are isomorphic.

The arguments are based on a strict topology which can be introduced in each Hilbert A-module V with respect to an essential ideal submodule X <= V.

Some applications will also be discussed.

Date received: May 14, 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-44.