Geometry of the unit sphere of aC∗-algebra and its dual

Abstract

In this study we exploit as a main tool a polar decomposition for linear functionals on operator algebras, introduced in 1958 by Sakai, to determine various types of extremal behavior in the unit spheres of C*-algebras and their duals. We discuss exposed points and complex extreme points as well as extreme points. Throughout this paper M, F, and A will be generic symbols for a von Neumann algebra, its predual, and a C*-algebra which may not have a unit, respectively. By a C*-algebra is meant a Banach *-algebra in which ||#*a;|| = \\x\\2 holds for all x. A von Neumann algebra is a C*-algebra of operators on a Hubert space which is closed in the weak operator topology and contains the identity operator. Each von Neumann algebra M is equivalent (as a Banach space) to the dual of the Banach space F of ultra-weakly continuous ( = normal) linear functionals on M. The space F is called the predual of M (and is unique). References for the preceding facts, as well as any others to follow concerning C* -algebras and von Neumann algebras, are the two monographs of Dixmier [3], [4], and the lecture notes of Sakai [13]. We will denote by w, s, uw, us the weak operator, strong operator, ultra-weak and ultra-strong topologies of M respectively, and n refers to the norm topology of a Banach