The Jacobson radical of a CSL algebra

Abstract

Extrapolating from Ringrose's characterization of the Jacobson radical of a nest algebra, Hopenwasser conjectured that the radical of a CSL algebra coincides with the Ringrose ideal (the closure of the union of zero diagonal elements with respect to finite sublattices). A general interpolation theorem is proved that reduces this conjecture for completely distributive lattices to a strictly combinatorial problem. This problem is solved for all width two lattices (with no restriction of complete distributivity), verifying the conjecture in this case. In [9] (cf. [3, Chapter 6]), Ringrose characterizes the Jacobson radical of a nest algebra. In [6], Hopenwasser described the appropriate generalization for reflexive algebras with commutative subspace lattice (CSL algebras), and verified his conjecture in a few special cases. The analogy with the nest case was pushed further in [8], clarifying the role of the carrier space. In this paper, we verify this conjecture for all width two CSL algebras. Moreover, we establish a framework for attacking the general problem and make significant progress in this direction. Recall that the Jacobson radical of a Banach algebra is the intersection of the kernels of all irreducible continuous representations. It also coincides with those elements T such that AT is quasinilpotent for every A in the algebra. Thus when 9 is a finite CSL, Alg(SF) is a finite matrix algebra with certain coefficients running over W(X) (and the rest 0). There is a unique contractive projection A, of the algebra onto the diagonal = Alg(S,) nAlg(F)* given by A_(T) = ZE ETE as E runs over the atoms of S . It is easy to see that the radical of Alg(Sr) is precisely the set of zero diagonal elements (T such that A_(T) = 0), which we denote Algo(SF). Indeed, this is a nilpotent ideal of order at most 1 I. Furthermore, it is easy to verify that Algo(F) n Alg(2') is an ideal in Alg(2') for all finite sublattices F of 2'; and indeed, it is a nilpotent ideal. Thus it is contained in the radical. It follows that the norm closure