$A$ is a $C^∗$-algebra and $P(A)$ is a set of projection of it. Assume that $A$ admits a strictly positive element $a$ such that $σ(a)-\{0\}$ is discrete.
I want to prove that $A$ admits an approximate unit $(p_n:\forall n \in N, p_n \in P(A))$
If I can prove that $C^∗(a)$ is *-isomorphic to $C_0(σ(a)-\{0\})$ , my problem is solved.
Because I know that $C_0(X)$ admits an approximate unit $(p_n:\forall n \in N, p_n \in P(A))$ if and only if $X=\cup X_n$ such that $\forall n,X_n $ is compact and open sets.
Q:Why can we say that $C^∗(a)$ is *-isomorphic to $C_0(σ(a)-\{0\})$?Is it true?Define *-isomorphism.
Your problem is solved by the following important fact which has been proved by Arens & Kadison 1n 1968 (see their paper: Pure state and approximate identities)
Theorem (Arense & Kadison). Let $a$ be an strictly positive in a C*-algebra $A$. The $\{a^{\frac{1}{n}}\}$ forms an approximate identity for $A$.
In your problem, you assumed $\sigma(a)$ is discrete. Therefore it is countable, namely $\{t_n\}_0^{\infty}$ where $t_0=0$. Now take $p_n=\sum_1^n \chi_{_{\{t_k\}}}(a)$ (since the spectrum is countable then $\chi_{_{\{t_k\}}}(a)$'s are all in $A$). The sequence of projections forms an approximate of identity.