tailieunhanh - Báo cáo toán học: "On factorial states of operator algebras"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên trạng thái thừa của các đại số nhà điều hành. | J. OPERATOR THEORY 15 1986 53-81 Copyright by INCREST 1986 ON FACTORIAL STATES OF OPERATOR ALGEBRAS. Ill R. J. ARCHBOLD and c. J. K. BATTY 1. INTRODUCTION In this paper we extend and clarify results from 6 8 19 concerning the pure state space P l and the factorial state space F z4 of a c -algebra A. Crucial to this programme is the notion of a primal ideal of a c -algebra defined in Section 3. Subsequently we investigate the associated notion of a weakly primal face of a compact convex set. To describe our results a convenient starting point is the following result A of Glimm Tomiyama and Takesaki 30 Theorem 2 in which PG4 denotes the set of pure states in the state space SG4 and the bar denotes w -closure ZA either A is one-dimensional A P A 2 S 1 J . . . or A is prime and antiliminal. This theorem has been split by the discovery in 6 8 of the following results B and C in which F T denotes the set of factorial states of A recall that p e S yf is factorial if the GNS representation 71 p gives rise to a commutant TtqfA which is a factor B F l 2 S i A is a prime c -algebra C . I or A contains an abelian ideal I such that AỊI is antiliminal. We shall denote by F X respectively Ff 1 Fj i Fln i the set of factorial states p such that TtọtẠỴ is type I respectively finite type I type II type III . Result D below is closely related to B and it was used in the proof of C see 8 D For any c -algebra A Ff A 2 F X . Finally result E was obtained in 6 as a corollary of B E If ker rv contains a prime ideal of A then p e F t . The converse is false. 54 . ARCHBOLD and c. í. K. BATTY Since it follows from D that Fị A 3 F 4 the possibility that FmC i F l was touched upon in 6 . Our first main result Theorem uses C to show that Fni .4 3 F A s A is antiliminal. Indeed this is analogous to C with Fjh X taking the role of PG4 . At the start of Section 3 we define the term primal ideal and then in Theorem we obtain the following generalization of B in which I is an ideal of the