The radical of a reflexive
operator algebra A whose lattice of invariant subspaces L is commutative is related
to the space of lattice homomorphisms of L onto {0,1}. To each such homomorphism
ϕ is associated a closed, two-sided ideal Aϕ contained in A. The intersection of the
Aϕ is contained in the radical; it is conjectured that equality always holds. The
conjecture is proven for a variety of special cases: countable direct sums of nest
algebras; finite direct sums of algebras which satisfy the conjecture; algebras whose
lattice of invariant subspaces is finite; algebras whose lattice of invariant
subspaces is isomorphic to the lattice of nonincreasing sequences with values in
N ∪{∞}.
