The linear span of projections in AH algebras and for inclusions of C*-algebras

A $C^{\ast }$-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 $A=\underset{}{\lim }(A_i,{\varphi }_i)$ has the LP property if and only if every real-valued continuous function on the spectrum of $A_i$ (as an element of $A_i$ 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 $C^{\ast }$-algebras $P \subset A$withafiniteWatataniIndex,ifafaithfulconditionalexpectation$E\colon A \rightarrow P$ 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 $C^{\ast }$-algebra with the LP property, $G$ a finite group and $\alpha$ an action of $G$ onto $\mathrm{A}\mathrm{u}\mathrm{t}(A)$. If $\alpha$ has the Rokhlin property in the sense of Izumi, then the fixed point algebra $A^G$ and the crossed product algebra $A \rtimes_\alpha G$ 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.