Unitization of stable $C^*$ algebras

Question

Suppose $A$ is a stable $C^*$-algebra,is the unitization of $A$ also stable?If not,can anyone show me some examples?

Answer 1

Note that for any $C^*$-algebras $A$ and $B$ and any tensor product $A\otimes B$, we have $A\otimes B$ is unital if and only if both $A$ and $B$ are unital. Since the algebra $\mathcal K$ of compact operators on a separable Hilbert space is non-unital, it follows that any stable $C^*$-algebra is non-unital, and therefore the unitization of any $C^*$-algebra is non-stable.