J. Aust. Math. Soc.
80 (2006), 297315

Operator algebras with a reduction property

James A. Gifford
Mathematical Sciences Institute
Australian National University
Canberra, ACT 0200
Australia
giffordj@maths.anu.edu.au



Abstract

Given a representation
of a Banach algebra
on a Hilbert space
,
is said to have the reduction property as an
module if every closed invariant subspace of
is complemented by a closed invariant subspace;
has the total reduction property if for every
representation ,
has the reduction property. We show that a
algebra has the total reduction property if and
only if all its representations are similar to
representations. The question of whether all
algebras have this property is the famous
`similarity problem' of Kadison. We conjecture
that nonselfadjoint operator algebras with the
total reduction property are always isomorphic to
algebras, and prove this result for operator
algebras consisting of compact operators.

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