  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 $A = \underrightarrow{\lim}(A_i,\phi_i)$ 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 $C^*$-algebras $P
\subset A$ with a finite Watatani Index, if a faithful conditional expectation
$E\colon A \rightarrow P$ 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 $C^*$-algebra with the
LP property, $G$ a finite group and $\alpha$ an action of $G$ onto
$\mathrm{Aut}(A)$. If $\alpha$ has the Rokhlin property in the sense of Izumi,
then the fixed point algebra $A^G$ and the crossed product algebra $A
\rtimes_\alpha G$ 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.
