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Operator algebras with a reduction property

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

James A. Gifford
James A. Gifford
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia, e-mail: giffordj@maths.anu.edu.au
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Abstract

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Given a representation θ: A → B(H) of a Banach algebra A on a Hilbert space H, H is said to have the reduction property as an A—module if every closed invariant subspace of H is complemented by a closed invariant subspace; A has the total reduction property if for every representation θ: A → B(H), H has the reduction property.

We show that a C*—algebra has the total reduction property if and only if all its representations are similar to *—representations. The question of whether all C*-algebras have this property is the famous ‘similarity problem’ of Kadison.

We conjecture that non-self-adjoint operator algebras with the total reduction property are always isomorphic to C*-algebras, and prove this result for operator algebras consisting of compact operators.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 46L07: Operator spaces and completely bounded maps
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 3 , June 2006 , pp. 297 - 315
Copyright
Copyright © Australian Mathematical Society 2006

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