Journal of Operator Theory

Volume 63, Issue 1, Winter 2010 pp. 115-128.

Geometric pre-ordering on $C^*$-algebras

Authors: Chi-Wai Leung (1), Chi-Keung Ng (2), and Ngai-Ching Wong (3)
Author institution: (1) Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
(2) Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
(3) Department of Applied Mathematics, National Sun Yat-sen University, and National Center for Theoretical Sciences, National Science Council, Kaohsiung, 804, Taiwan

Summary: It has been a successful practice to define a canonical pre-ordering on a normed space using the inclusion of faces of its closed dual unit ball. This pre-ordering reflects some geometric property in a natural way. In this article, we will give an algebraic description of this pre-ordering in the case of complex $C^*$-algebras as well as that of their self-adjoint parts. In developing our theory we introduce the \textit{essential support} of an element, which is closely related to the notion of peak projections studied recently by Blecher and Hay. As applications, we give some interesting facts about weak*-closed faces, and will identify the quasi-maximal elements and the quasi-minimal elements with respects to this pre-ordering. They are closely related to the extreme points and the smooth points of the unit sphere of the $C^*$-algebra.

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