The Linear Span of Projections in AH Algebras and for Inclusions of C * -Algebras

Abstract

In the first part of this paper, we show that an AH algebra $A=\underrightarrow{\lim }(A_i,{\varphi }_i)$ has the LP property if and only if every element of the centre of $A_i$ 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 in the sense of Bratteli and Elliott. The second contribution of this paper is that for an inclusion of unital $C^{*}$-algebras $P\subset A$ with a finite Watatani index, if a faithful conditional expectation $E:A\to P$ has the Rokhlin property in the sense of Kodaka et al., then $P$ has the LP property under the condition that$A$ has the LP property. As an application, let $A$ be a simple unital $C^{*}$-algebra with the LP property, $\alpha$ an action of a finite group $G$ onto $\text{Aut}(A)$. If $\alpha$ has the Rokhlin property in the sense of Izumi, then the fixed point algebra $A^G$ and the crossed product algebra $\mathrm{A\hspace{0.17em}}{⋊}_{\alpha }\mathrm{\hspace{0.17em}G}$ have the LP property. We also point out that there is a symmetry on the CAR algebra such that its fixed point algebra does not have the LP property.