Is it true that $I \otimes J$ is ideal of $A \otimes B$?

Question

Let $A$ and $B$ be $C^{\ast}-$ algebras and $A \otimes B$ denotes minimal(spatial) tensor product. Let $I$ and $J$ be ideals of $A$ and $B$ respectively. Is it true that

Is it true that $I \otimes J$ is ideal of $A \otimes B$?

It seems true but I cannot see the proof. Any ideas?

Answer 1

$I\otimes J$ can be treated as a C* subalgebra in $A\otimes B$ in a natural way - see remark 6.5.1. in "C*-algebras and operator theory" Gerard J. Murphy.

Using this identification it is trivial that $I\otimes J$ will be an ideal - one just needs to look at simple tensors and argue in the general case by density.