The radical of a reflexive operator algebra

Abstract

The radical of a reflexive operator algebra % whose lattice of invariant subspaces 2 is commutative is related to the space of lattice homomorphisms of 2 onto {0,1}. To each such homomorphism φ is associated a closed, two-sided ideal Wψ contained in H. The intersection of the ^ 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 U {°°}. 1* Introduction* This paper studies the radical of a certain class of non-self-ad joint operator algebras. Given an algebra 21 and a lattice S of orthogonal projections acting on a separable Hubert space φ, we use the standard notations, Sat 21 and SttgS, to denote, respectively, the lattice of all projections invariant under Sί and the algebra of all (bounded) operators which leave invariant each projection of 2. 2t and 2 are said to be reflexive if 2ί = Stlg Sat 2C and S = Sat 2CIg S, respectively. The algebras which we study are reflexive algebras which contain a maximal abelian self-adjoint algebra (m.a.s.a.). A commutative subspace lattice is a lattice of pairwise commuting, orthogonal projections on $ which contains 0 and 1 and which is closed in the strong operator topology. It follows automatically that a commutative subspace lattice is a complete lattice. If 2C is an operator algebra containing a m.a.s.a., then Sat SI is a commutative subspace lattice. Every commutative subspace lattice, S, is reflexive ([1], p. 468), and SίlgS is a reflexive algebra which contains a m.a.s.a. Henceforth, all lattices of projections in this paper will be commutative subspace lattices and all algebras will be reflexive algebras which contain a m.a.s.a. An incisive study of these lattices and algebras by Arveson is found in [1]. At least in certain special cases, the radical of a reflexive algebra, 2ί, containing a m.a.s.a. can be described in terms of the set of lattice homomorphisms from S = Sαt2t onto {0,1}. To each such homomorphism φ we shall associate a closed two-sided ideal %φ in St. The radical, % of Sf is equal to the intersection of these ideals. It appears reasonable to conjecture that this equality holds for all