Mapping on Standard C*algebras Preserving Comparison for Positive Elements

Abstract:

Let H be a complex Hilbert space,AH the standard C*algebra on H,and AH+ the set of all positive elements in AH.In 1977,Cuntz introduced a comparison in the set of all positive elements in a C*algebra.Let A be a C*algebra.For A,B∈A+,we write A≤B if there exists an element X∈A such that A=XBX*.In this paper,we characterize all weakly continuous semilinear surjective maps on AH+ which preserves the positive elements comparison in both directions.