Existence of spectral well-behaved ^*-representations

Abstract

There are several cases, where an m^*-seminorm _p is defined on a ^*-subalgebra of a given ^*-algebra A. This may lead to the construction of an unbounded ^*-representation of A. Such m^*-seminorms are called unbounded. Given an unbounded m^*-seminorm_p of a ^*-algebra A, the concept of a _p-spectral ^*-representation of A is introduced and studied in connection to well-behaved ^*-representations. More precisely, the existence of (p-) spectral well-behaved ^*-representations is investigated on ^*-algebras and locally convex ^*-algebras in terms of certain properties of Ptak function, closely related to hermiticity and C^*-spectrality of the ^*-subalgebras on which this function is defined. Various examples in diverse classes of locally convex algebras illuminate the elaborated results.