Ideals and Quotients of C*-Algebras

Complete the proof of the following proposition: if $X$ is compact and $I$ is a closed ideal of $C(X)$, then there is a closed subset $F$ of $X$ such that $I = \{ f \in C(X) : f(x) = 0 \text{ for all } x \text{ in } F\}$. Moreover, $C(X)/I$ is isometrically isomorphic to $C(F)$.

Show that $M_n(\mathbb {C})$ has no nontrivial ideals. Find all of the left ideals.

If $\alpha$ is an infinite cardinal number, let $I_\alpha = \{ A \in \mathcal{B}(\mathcal{H}) : \dim \operatorname {cl}(\operatorname {ran} A) \leq \alpha \}$. Show that $I_\alpha$ is a closed ideal in $\mathcal{B}(\mathcal{H})$.

Let $S$ be the unilateral shift on $l^2$. Show that $C^*(S) \supseteq \mathcal{B}_0(l^2)$ and $C^*(S)/\mathcal{B}_0(l^2)$ is abelian. Show that the maximal ideal space of $C^*(S)/\mathcal{B}_0(l^2)$ is homeomorphic to $\partial \mathbb {D}$.

If $V$ is the Volterra operator on $L^2(0,1)$, show that $C^*(V) = \mathbb {C} + \mathcal{B}_0(L^2(0,1))$.

If $\mathcal{A}$ is a $C^*$-algebra, $I$ is a closed ideal of $\mathcal{A}$, and $\mathcal{B}$ is a $C^*$-subalgebra of $\mathcal{A}$, show that the $C^*$-algebra generated by $I \cup \mathcal{B}$ is $I + \mathcal{B}$.

If $\mathcal{A}$ is a $C^*$-algebra and $I$ and $J$ are closed ideals in $\mathcal{A}$, show that $I + J$ is a closed ideal of $\mathcal{A}$.