TAILIEUCHUNG - Báo cáo toán học: "Simplexes of states of C*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Simplexes của bang C *- đại số. | J. operator theory 4 1980 3-23 Copyright by INCREST 1980 SIMPLEXES OF STATES OF C -ALGEBRAS c. J. K. BATTY 1. INTRODUCTION The state space S A of a unital c -algebra A becomes a compact convex set when equipped with the weak topology and this fact has been the motivation for much of the general theory of compact convex subsets of locally convex spaces 1 . The outstanding problem of which compact convex sets are affinely homeomorphic to state spaces of c -algebras has recently been solved by Alfsen and Shultz 3 but for Choquet simplexes the solution has long been known a simplex is the state space of a c -algebra if and only if it is a Bauer simplex . its extreme boundary is compact ill which case the c -algebra is commutative. However other simplexes may occur as closed faces of state spaces. Among the properties of state spaces observed by Alfsen and Shultz and inherited by their faces is the 3-ball property namely that the face generated by two pure states of a c -algebra is either their convex hull or is affinely isomorphic to a 3-dimensional Euclidean ball. In the reverse direction it is immediately apparent that any simplex has the 1-ball property . the convex hull of any two extreme points is a face. In- many physical applications a restricted class of states of the c -algebra is of Special interest and attempts to decompose these into extremal states involve determining whether the class forms a simplex. For example if A is the c -algebra of observables of a quantum system the symmetries of the system are represented by a group G of -automorphisms of A and the space SC A of G-invariant states merits special attention. Various notions of asymptotic abelianness were introduced which ensure that SG A forms a simplex 10 11 12 13 18 24 25 31 Lanford and Ruelle 19 25 26 considered G-invariant states cp with associated covariant representations p wv which are G-abelian in the sense that 7Tv 4 restricts to an abelian von Neumann algebra on the space of .

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