Selfadjoint representations of polynomial algebras

Abstract

In this paper we will investigate the selfadjointness of unbounded ∗ ^{\ast } -representations of the polynomial algebra. In particular, it is shown that the notion of selfadjoint representation is equivalent to that of standard representation in the case of the polynomial algebra generated by one hermitian element. Although the notion of standardness implies that of selfadjointness, the converse is not true in general. Therefore, we consider under what conditions a ∗ ^{\ast } -representation is standard.