Is the stabilization of a simple $C^*$-algebra simple?

Question

Recall that a $C^*$-algebra is called simple if it has no non-trivial closed two-sided ideals.

Let $\mathcal{K}$ be the $C^*$-algebra of compact operators on a separable infinite-dimensional Hilbert space $H$.

Question: It true that a $C^*$-algebra $A$ is simple if and only if the tensor product $A\otimes\mathcal{K}$ is simple?

Thoughts: One direction is obvious, since if $I$ is a non-trivial ideal in $A$, then $I\otimes\mathcal{K}$ is an ideal in $A\otimes\mathcal{K}$. What can be said about the converse?

Answer 1

Let $A,B$ be $C^*$-algebras. By a result of Takesaki, $A\otimes B$ (minimal tensor product) is simple iff $A$ and $B$ are simple (reference: chapter IV in Takesaki's first book).

Since the algebra of compact operators is simple, it follows that the answer to your question is positive.