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Journal of the London Mathematical Society Advance Access published online on December 14, 2007
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm099
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© 2007 London Mathematical Society

On abelian, triangularizable, total reduction algebras

L. W. Marcoux

Department of Pure Mathematics
University of Waterloo
Waterloo
Ontario
Canada N2L 3G1

Suppose that h is a complex Hilbert space and that Formula (h) denotes the bounded linear operators on h. In this paper we show that if Formula {subseteq} Formula (h) is an abelian algebra with elements that admit representations under which they have a spanning set of eigenvectors, and if Formula has the property that given any bounded representation {varrho}: Formula ->Formula (h{varrho}) of Formula on a Hilbert space h{varrho}, every invariant subspace of {varrho}(Formula ) is topologically complemented by an invariant subspace of {varrho}(Formula ), then Formula is similar to an abelian C*-algebra. From this it follows that any amenable Banach algebra of triangular operators on a separable Hilbert space is similar to an abelian C*-algebra.

Research supported in part by NSERC (Canada).

2000 Mathematics Subject Classification 46J40 (primary), 47A25, 47A66, 47C05 (secondary).

Received July 5, 2006; revised July 13, 2007;
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