The Cuntz Comparison in the Standard C*-Algebra

Abstract

The Cuntz comparison, introduced by Cuntz in early 1978, associates each ${\text{C}}^{*}$-algebra with an abelian semigroup which is an invariant for the classification of the nuclear ${\text{C}}^{*}$-algebras and called the Cuntz semigroup. In this paper, we study the Cuntz comparison in the standard ${\text{C}}^{*}$-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.