9 - *-Algebras pages 919 to 1032

Summary

A topologically cyclic *-representation of a *-algebra A is determined up to unitary equivalence by a certain type of linear functional on A. We will call the linear functional that are associated with topologically cyclic *-representations in this special way, representable positive linear junctionals. From §9.4.2 to §9.4.16 we give a construction, due to Israel Moiseevič Gelfand and Mark Aronovič Naǐmark [1943], of a *-representation T^ω from each representable positive linear functional ω. This construction was further developed by Irving E. Segal [1947a]. If ω is associated with the topologically cyclic *-representation T, then T^ω is unitarily equivalent to T. Thus our construction gives a representative in each unitary equivalence class of topologically cyclic *-representations. In Section 9.2 we showed that each *-representation is the Hilbert sum of a trivial *-representation and an essential *-representation, and that each essential *-representation is unitarily equivalent to a Hilbert sum of topologically cyclic *-representations. Thus the Gelfand-Naimark construction and the Hilbert sum construction together give a representative in each equivalence class of *-representations.