Approximation of unity by projectors

Question

Let $A$ be a $\sigma$-unital $C^*$-algebra and $A_s:=A\otimes K$ its stabilization (where $K$ is the algebra of compact operators on a separable Hilbert space). Is it true that there exist an approximation of unity $P_n\in A_s$ with $P^*_n=P_n=P_n^2$, in general?

Answer 1

What if $A=C_0([0,1))$, the continuous functions $f:[0,1]\rightarrow\mathbb C$ with $f(1)=0$? Then $A\otimes K = C_0([0,1), K)$ the space of norm continuous $f:[0,1]\rightarrow K$ with $f(1)=0$.

Then, if $f=f^*=f^2$ then $f(s)=f(s)^* = f(s)^2$ for each $s\in [0,1]$. Thus $f(s)$ is a projection for each $s$, so $\|f(s)\|=0$ or $1$ for each $s$. That $f$ is continuous and $f(1)=0$ forces $f=0$. So $A_s$ has no (non-zero) projections in this case.