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Abstract and Applied Analysis
Volume 2014, Article ID 623520, 4 pages
http://dx.doi.org/10.1155/2014/623520
Research Article

The Cuntz Comparison in the Standard -Algebra

1Department of Mathematics, Tongji University, Shanghai 200092, China
2Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
3School of Science, Shanghai Institute of Technology, Shanghai 201418, China

Received 13 March 2014; Accepted 21 May 2014; Published 5 June 2014

Academic Editor: Dumitru Motreanu

Copyright © 2014 Xiaochun Fang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The Cuntz comparison, introduced by Cuntz in early 1978, associates each -algebra with an abelian semigroup which is an invariant for the classification of the nuclear -algebras and called the Cuntz semigroup. In this paper, we study the Cuntz comparison in the standard -algebra. We characterize the Cuntz comparison in terms of the dimension of the operator range. Also, we consider the structure of the semilinear map which preserves the Cuntz comparison.