Let ~¢g be a complex, not necessarily separable, Hilbert space and let ~( g ) denote the algebra of all (bounded, linear) operators on Jr. A subalgebra 9.I of ~( J f ) is called reductive if every invariant (closed) subspace of ovg reduces 91, i.e., its orthogonal complement is also invariant under 91.1. We do not require, as part of the definition, that 91 be closed in any topology; nor do we hypothesize the existence of a unit in 91. The reductive algebra problem is to determine whether all weakly closed reductive algebras are necessarily self-adjoint [7, p. 167]. Although the problem is still open, several stronger hypotheses have been found to imply that such an algebra is self-adjoint. For a detailed discussion of the subject see [7].Nordgren and Rosenthal [-5] proved that if a reductive algebra 9.1 is weakly closed and if the set of finite-rank operators in 91.1 is separating (i.e., given a nonzero x in Jig, there exists a finite-rank operator F in 91.I with Fx + 0) then 91 is selfadjoint. After Lomonosov's result [3] it was possible to prove a stronger theorem replacing "finite-rank" by "compact" in the above theorem [-10]. (See also [1] and [8].) It is the purpose of the present paper to extend the Nordgren-Rosenthal result in another, algebraic, direction. The role of finite-rank operators will be replaced by that of minimal one-sided ideals, but there is another added feature: our main result does not assume any topological closure on 91. Thus we characterize all reductive algebras whose left (or right) socles are separating. (Recall that the left socle is the sum of all minimal left ideals.) Actually, the two socles coincide for reductive algebras. (See Lemma 1 below.)If A is an operator on ~f~ and 09 is a cardinal number, we denote by A ~') the direct sum of co copies of A. If 9i is a subalgebra of N(Jg), 91~o,) will denote the algebra {A C'°) :A~92[}. If o~ is a transitive subalgebra of ~(~,ug), i.e., o~ has no invariant subspaces other than {0} and ovg, and if ~ consists of finite-rank operators, then ~o,) is a reductive algebra and its weak closure is ~( ) f ) ~°'). [This follows from three facts: the weak closure of o~ is ~(~) [4], ~(o'¢g) ~°' is reductive, even self-adjoint, and an algebra has the same invariant subspaces as its weak
