is said to have the reduction property if every closed subspace M of H

which is invariant under the action of all elements of A is complemented

by another subspace N which is also invariant for A. The algebra is said

to have the total reduction property if every continuous representation of

A acting on a second Hilbert space K also has the reduction property.

It has been conjectured that every norm-closed subalgebra of Hilbert space

operators with the total reduction property is similar to a

C<sup>*</sup>-algebra of operators. In this talk, we shall examine this

conjecture in the context of abelian algebras of operators, and relate the

total reduction property to amenability, another well-studied property of

Banach algebras.