A norm-closed algebra A of bounded linear operators on a Hilbert space H
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.