Existence of spectral well-behaved ∗-representations

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 Pták 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.