Uniform topologies on enveloping algebras

Abstract

Let G be a Lie group with Lie algebra g and E ( G) the unversal enveloping algebra of g realized as the algebra of left-invariant differential operators on G. It is proved that the uniform topology on E ( G), i.e., the topology of uniform convergence on weakly bounded sets of vector states, coincides with the strongest locally convex topology on E ( G). This implies that each linear functional on E ( G) is a linear combination of strictly positive functionals.