stably finite $C^*$ algebras

Question

I saw a conlusion:If $A$ is simple $C^*$ algebra,then $A\otimes \Bbb K$ contains no infinite projections?

If $A$ is simple, $A\otimes \Bbb K $ contains no infinite projections,can we conclude that $A$ is stably finite?

Answer 1

Take any simple, purely infinite C$^*$-algebra; for instance, $\mathcal O_2$. Then $A\otimes\mathbb K$ contains lots of infinite projetions: $1\otimes p$ is infinite for any projection $p$.

The second question is easily answered if you notice that $M_n(A)=A\otimes M_n(\mathbb C)$ embeds in $A\otimes \mathbb K$.