$C^*$-ideals in tensor products

Question

This post contains several basic questions about ideals in tensor products of $C^*$-algebras, motivated by the comments in this post.

Let $A$ be a $C^*$-algebra, and let $\mathcal{K}$ denote the compact operators on a Hilbert space $H$. I will use the term "ideal" to mean "closed, two-sided ideal".

Let $I$ be a non-trivial ideal of $A$, i.e. $0\neq I\neq A$. Then $I\otimes\mathcal{K}$ is an ideal of $A\otimes\mathcal{K}$.

Question 1: Is $I\otimes\mathcal{K}$ a non-trivial ideal?

Question 2: Is every ideal of $A\otimes\mathcal{K}$ of the form $I\otimes\mathcal{K}$? Why?

Thoughts: For Q1, one could think of the inclusion $I\otimes\mathcal{K}\subseteq A\otimes\mathcal{K}$ as a direct limit of matrix algebra inclusions $M_n(I)\subseteq M_n(A)$. For each $n$, it is certainly true that $M_n(I)$ is neither zero nor all of $M_n(A)$. But I'm not sure why the proper inclusion still holds after taking the limit.

Also, I wonder if this is true more generally:

Question 3: If we replace $\mathcal{K}$ with an arbitrary $C^*$-algebra $B$, then is the analogue of Q1 still true? More precisely, if we take spatial tensor products, is $I\otimes B$ still a non-trivial ideal of $A\otimes B$? Does the answer change if we take other tensor products?

Answer 1

1. It is always non-trivial. Since the ideal $J$ is a closed subspace of $A$, by Hahn-Banach we can find $\varphi\in A^*$ with $\|\varphi\|=1$, $\varphi|_J=0$ and $\varphi(a)=1$. Fix $k_0\in K(H)$ and fix a state $\psi\in K(H)^*$ with $\psi(k_0)=1$ (we can take $\psi$ to a point state). Now consider an element $\sum_j b_j\otimes k_j\in J\otimes K(H)$. We have $$ \|a\otimes k_0-\sum_jb_j\otimes k_j\|\geq \Big|(\varphi\otimes \psi)\big(a\otimes k_0-\sum_jb_j\otimes k_j\big)\Big|=\varphi(a)\psi(k_0)=1. $$ So $a\otimes k_0$ is at distance 1 from every elements in a dense subset of $J\otimes K(H)$, which shows that $a\otimes k_0\not\in J\otimes K(H)$.

2. I think the answer is yes, but I would have to think how to prove it.

3. The argument in 1 applies in general, as long as $J$ is a closed subspace of $A$. The only nontrivial part of the proof is whether $\varphi\otimes \psi$ defines a bounded linear functional on the tensor product. This can be done for states and any C$^*$-tensor norm by Proposition 3.4.7 in Brown-Ozawa. As any bounded linear functional is a linear combination of states (see for instance Theorem 3.3.10 in Murphy), this works for all bounded functionals.