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? 



Approximation of unity by projectors
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.


