On commutatively dominated Op∗-algebras with Fréchet domains

Abstract

Suppose that the maximal Op ∗-algebra L +( D) on a Fréchet domain D contains a sequence of strongly commuting essentially self-adjoint operators which is cofinal in the set of all hermitian elements of L +( D). Then each positive linear functional on L +( D) decomposes in a unique way into a sum of an ultraweakly continuous positive linear functional, a uniformly continuous positive linear functional which vanishes on the finite rank operators, and a positive linear functional which vanishes on the dense subspace of the “very continuous” operators. For bounded linear functionals on certain subspaces of the completion L of L +( D), similar results are satisfied. These results are connected with properties of the uniform topology and the associated bornological topology of L .