2012/10/19 by Dinh Trung Hoa, Ho Minh Toan, Hoa, Dinh Trung +3
Mathematics · #46L35 #46L55 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1210.5426
openalex publication_date 2012/10/19 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28
A C^*-algebra is said to have the LP property if the linear span of projections is dense in a given algebra. In the first part of this paper, we show that an AH algebra A = \underrightarrowlim(Ai,ϕi) has the LP property if and only if every real-valued continuous function on the spectrum of Ai (as an element of Ai via the non-unital embedding) belongs to the closure of the linear span of projections in A. As a consequence, a diagonal AH-algebra has the LP property if it has small eigenvalue variation. The second contribution of this paper is that for an inclusion of unital C^*-algebras P ⊂ A with a finite Watatani Index, if a faithful conditional expectation E\colon A → P has the Rokhlin property in the sense of Osaka and Teruya, then P has the LP property under the condition A has the LP property. As an application, let A be a simple unital C^*-algebra with the LP property, G a finite group and α an action of G onto Aut(A). If α has the Rokhlin property in the sense of Izumi, then the fixed point algebra AG and the crossed product algebra A \rtimesαG have the LP property. We also point out that there is a symmetry on CAR algebra, which is constructed by Elliott, such that its fixed point algebra does not have the LP property.