J. Aust. Math. Soc.  80 (2006), 297-315
Operator algebras with a reduction property

James A. Gifford
  Mathematical Sciences Institute
  Australian National University
  Canberra, ACT 0200

Given a representation $\theta:\mathcal{A} \to\mathcal{B}(H)$ of a Banach algebra $\mathcal{A}$ on a Hilbert space $H$, $H$ is said to have the reduction property as an $\mathcal{A}$-module if every closed invariant subspace of $H$ is complemented by a closed invariant subspace; $\mathcal{A} $ has the total reduction property if for every representation $\theta:\mathcal{A}\to\mathcal{B}(H)$, $H$ 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.
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