Operator algebras with a reduction property

• Published in: Journal of the Australian Mathematical Society (2001) Vol. 80; no. 3; pp. 297 - 315
• Main Author: Gifford, James A.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.06.2006
• Subjects: 46L07; Algebra; Banach spaces; Hilbert space; Invariants; Linear operators; Operators (mathematics); primary 46L05; Reduction; Representations; primary 46L05; 46L07
• ISSN: 1446-7887; 1446-8107
• DOI: 10.1017/S1446788700014026

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.