Partial *-algebras of Distributions

Abstract

The problem of multiplying elements of the conjugate dual of certain kind of commutative generalized Hilbert algebras, which are dense in the set of $C^{\infty }$-vectors of a self-adjoint operator, is considered in the framework of the so-called duality method. The multiplication is defined by identifying each distribution with a multiplication operator acting on the natural rigged Hilbert space. Certain spaces, that are an abstract version of the Bessel potential spaces, are used to factorize the product.