Unitary equivalence of unbounded *-representations of *-algebras

Abstract

In this paper the unitary equivalence of unbounded *-representations of *-algebras is investigated. It is shown that if closed *-representations π 1 and π 2 of a *-algebra [Ascr ] satisfy a certain density condition for the intertwining spaces [Jscr ](π 1 , π 2 ) and [Jscr ](π 2 , π 1 ), then a *-isomorphism Φ between the O *-algebras π 1 ([Ascr ]) and π 2 ([Ascr ]) is defined by Φ(π 1 ( x ))=π 2 ( x ), x ∈[Ascr ] and it induces a *-isomorphism Φ¯, between the von Neumann algebras (π 1 ([Ascr ]) ′ w ) ′ and (π 2 ([Ascr ]) ′ w ) ′ , and further if Φ¯, is spatial (that is, it is unitarily implemented), then π 1 and π 2 are unitarily equivalent.