Geometric pre-ordering on C*-algebras

Chi-Wai Leung Chi-Keung Ng Ngai-Ching Wong

Functional Analysis mathscidoc:1910.43716

Journal of Operator Theory, 115-128, 2010.1
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 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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  title={Geometric pre-ordering on C*-algebras},
  author={Chi-Wai Leung, Chi-Keung Ng, and Ngai-Ching Wong},
  booktitle={Journal of Operator Theory},
Chi-Wai Leung, Chi-Keung Ng, and Ngai-Ching Wong. Geometric pre-ordering on C*-algebras. 2010. In Journal of Operator Theory. pp.115-128.
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