Is $A$ a hereditary subalgebra of $A\otimes\mathcal{K}$

Question

Let $A$ be a $C^\ast$-algebra, $\mathcal{K}$ the algebra of all compact operators on a separable Hilbert space. Is $A$ (isomorphic to) a hereditary subalgebra of $A\otimes\mathcal{K}$?

Thanks.

Answer 1

If $A$ is $\sigma$-unital you can do the following. Let $h_A \in A$ be a strictly positive element. Then you can consider the corner $$ \overline{h(A \otimes \mathcal K)h} \ , $$ where $h = h_A \otimes e_{11}$.