On Dual Algebras And Their Preduals.

Abstract

Let H be a complex, separable Hilbert space, let L(H) be the algebra of bounded operators on H, and let (tau) be a topology which makes L(H) into a locally convex topological vector space. A dual algebra A is an ultraweakly closed, unital subalgebra of L(H). Let Q(,A), be the space of ultraweakly continuous forms on A. Then A is the dual of Q(,A). We study the structure of Q(,A), the predual of A. In Chapter I we consider (tau)-continuous forms on a linear submanifold M in L(H). A form (phi) on M is a vector form if there exist x,y in H such that (phi)(A) = (Ax,y), A (ELEM) M. M is (tau)-vectorial if every (tau)-continuous form on M is a vector form. We give necessary and sufficient conditions for t-vectoriality in terms of reflexivity. For (tau)-vectorial algebras we obtain some invariant subspace theorems and a dilation theory. In Chapter II we attempt to generalize a result which characterizes the ultraweakly vectorial von Neumann algebras. We obtain new information about the predual of a von Neumann algebra with properly infinite commutant. We study n-fold ampliations of dual algebras. In Chaper III we exhibit a classification of dual algebras where the classes are analogous to the classes of contraction C(,0(.)), C(,1(.)), etc. We give several examples of the types of dual algebras.