K- theory of stably projectionless C* algebras

Question

could anyone give me an example of a stably projectionless C*-algebra with non-zero $K_0$ group?

Answer 1

Indeed, $C_0(\mathbb R^2)$ provides such an example. Based on the answer to your previous question, $K_0(C_0(\mathbb R^2)=\mathbb Z$ is nontrivial. To show that $C_0(\mathbb R^2)$ is stably projectionless, consider any map $f:\mathbb R^2\to M_n(\mathbb C)$, vanishing at infinity, with $f(x)$ a projection for all $x\in\mathbb R^2$. Since any nontrivial projection has norm $1$, and $\mathbb R^2$ is connected, it follows that $f=0$, and the result follows.