Preservation of meets by a *-homomorphism between $C^*$-algebras

Question

Let $\varphi :A\to B$ be a *-homomorphism between $C^*$-algebras. If I is a closed ideal of A, then one can consider $Id(\varphi (I))$, the closed ideal of B generated by the image $\varphi (I).$ Is it true that the operation $I\mapsto Id(\varphi (I))$ preserves meets? In other words, if $I$ and $J$ are closed ideals, does the following equality hold? $$Id(\varphi (I\cap J))=Id (\varphi (I))\cap Id (\varphi (J))$$ If not, under what conditions is it true?

Answer 1

The inclusion $$\tag1Id(\varphi (I\cap J))\subset Id (\varphi (I))\cap Id (\varphi (J))$$ always holds. The converse can fail. For instance, let $A=\mathbb C^2$, $B=M_2(\mathbb C)$, and $$ \varphi(a,b)=\begin{bmatrix} a&0\\0&b\end{bmatrix}. $$ Put $I=\mathbb C\oplus 0$, and $J=0\oplus \mathbb C$. Then $I\cap J=\{0\}$. So $$Id(\varphi (I\cap J))=\{0\},$$ while, as $B$ is simple, $$ Id(\varphi (I))=Id(\varphi (J))=M_2(\mathbb C). $$ The condition is obviously true when $\varphi$ is an isomorphism; I cannot think of another condition, as it could depend both on $\varphi$ and $A,B$.