Dense m-convex Frechet Subalgebras of Operator Algebra Crossed Products by Lie Groups

Abstract

Let A be a dense Frechet *-subalgebra of a C*-algebra B. (We do not require Frechet algebras to be m-convex.) Let G be a Lie group, not necessarily con- nected, which acts on both $A$ and B by *-automorphisms, and let \s be a sub- multiplicative function from G to the nonnegative real numbers. If \s and the action of G on A satisfy certain simple properties, we define a dense Frechet *-subalgebra G\rtimes^{\s} A of the crossed product L^{1}(G, B). Our algebra consists of differentiable A-valued functions on G, rapidly vanishing in \s. We give conditions on the action of G on A which imply the m-convexity of the dense subalgebra G\rtimes^{\s}A. A locally convex algebra is said to be m-con- vex if there is a family of submultiplicative seminorms for the topology of the algebra. The property of m-convexity is important for a Frechet algebra, and is useful in modern operator theory. If G acts as a transformation group on a manifold M, we develop a class of dense subalgebras for the crossed product L^{1}(G, C_{0}(M)), where C_{0}(M) denotes the continuous functions on M vanishing at infinity with the sup norm topology.We define Schwartz functions S(M) on M, which are differentiable with respect to some group action on M, and are rapidly vanishing with respect to some scale on M. We then form a dense m-convex Frechet *-subalgebra G\rtimes^ {\s} S(M) of rapidly vanishing, G-differentiable functions from G to S(M). If the reciprocal of \s is in L^{p}(G) for some p, we prove that our group algebras S^{\s}(G) are nuclear Frechet spaces, and that G\rtimes^{\s}A is the projective completion S^{\s}(G) \otimes A.