Existence of well-behaved *-representations of locally convex *-algebras

Abstract

The following problems are investigated: (1) The existence of well-behaved *-representations on a *-algebra A equipped with an unbounded m^*-seminorm p, in terms of non-zero p-continuous representable (positive) linear functionals on the domain A(p) of p . (2) The existence of well-behaved *-representations of a locally convex *-algebra A, in terms of non-zero unbounded C ^*-seminorms on A with domain the *-subalgebra A_b generated by the hermitian part of the Allan bounded set A₀ of A. (3) The existence of faithful well-behaved *-representations of a locally convex *-algebra A, in terms of the so-called unbounded Gel'fand-Naimark C ^*-seminorm on A.