Bounded vectors for unbounded representations and standard representations of polynomial algebras

Abstract

Let A be a unital commutative ^*-algebra. Let π be a hermitian representation of A into (not necessarily bounded) Hilbert space operators. Analytic vectors and bounded vectors for π are investigated; and are used to show that π is a direct sum of bounded (operator) representations iff π admits a core consisting of bounded vectors. This, in turn, is used to show that if A is either of the polynomial algebras ζ(x) or ζ(x, y) in one or two commuting hermitian generators then π is standard iff π is a direct sum of bounded representations. Various selfadjointness and standardness criteria for representations of these polynomial algebras are developed, highlighting the difference between the representation theory of these two algebras, and supplementing known results.