On Reductive and Transitive Operator Algebras.

Abstract

Let H be a separable, infinite dimensional, complex Hilbert space. Let L(H) denote the algebra of all operators on H. A subspace M is invariant for the operator A if A MCM. A subalgebra A of L(H) is said to be transitive if the only invariant subspaces for A are {0} and H. A subalgebra W of L(H) is said to be reductive if every invariant subspace of W is reducing. The following problems are well known in operator theory. The Transitive Algebra Problem: If U is a transitive subalgebra of L(H), must U be strongly dense in L(H)? The Reductive Algebra Problem: If W is a reductive algebra, must W(,s), the closure of W in the strong operator topology, be a von Neumann algebra? In this paper we will focus on the above problems, and we will obtain several partial solutions for the transitive and reductive algebra problems.