*-Representations of Partial *-Algebras

Abstract

This chapter is devoted to *-representations of partial *-algebras. We introduce in Section 7.1 the notions of closed, fully closed, self-adjoint and integrable *-representations. In Section 7.2, the intertwining spaces of two *-representations of a partial *-algebra are defined and investigated, and using them we define the induced extensions of a *-representation. Section 7.3 deals with vector representations for a *-representation of a partial *-algebra, which are the appropriate generalization to a *-representation of the notion of generalized vectors described in Chapter 5. Regular and singular vector representations are defined and characterized by the properties of the commutant, and an arbitrary vector representation is decomposed into a regular part and a singular part. Section 7.4 deals with *-subrepresentations of a *-representation of a partial *-algebra. Let π be a fully closed *-representation of a partial *-algebra A. A *-subrepresentation π M of π is defined by any reducing subspace M of the domain D(π) of π, but the projection E M from the Hilbert space H π onto the closure does not necessarily belong to the quasi-commutant C qw(π) of π. If E M ∈ C qw(π), then another *-subrepresentation of π is defined and π M ⊂ , but in general. Thus one proceeds to investigate the relation between π M and and their self-adjointness. Section 7.5 deals with the unitary equivalence of *-representations of a partial *-algebra and the spatiality of *-automorphisms of a partial O*-algebra.