Show that the ideal is closed.

Question

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

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

It Is clear that $I \otimes J$ is ideal but how to prove that this ideal is closed?

Answer 1

In general, the minimal tensor product $A \otimes B$ is defined as an abstract Banach space completion of $A \odot B$. Similarly, we can form the the minimal tensor product $I \otimes J$.

There is a canonical $*$-homomorphism $$I \otimes J \to A \otimes B.$$ Using this $*$-homomorphism, we can identify $I \otimes J$ as a subset of $A \otimes B$. The image of a $*$-homomorphism is automatically closed, so actually your question is a matter of definition: the answer to your question is yes, by the way we define $I \otimes J$ as a subset of $A \otimes B$.