Ideals and Quotients

Let $\mathcal{A}$ be a Banach algebra and let $\mathcal{L}$ be all of the closed left ideals in $\mathcal{A}$. If $I_1, I_2 \in \mathcal{L}$, define $I_1 \vee I_2 \equiv \text{cl}(I_1 + I_2)$ and $I_1 \wedge I_2 = I_1 \cap I_2$. Show that with these definitions $\mathcal{L}$ is a complete lattice with a largest and a smallest element.

Let $X$ be locally compact. For every open subset $U$ of $X$, let $I(U) = \left\{ \phi \in C_0(X) : \phi = 0 \text{ on } X \setminus U\right\}$. Show that $U \mapsto I(U)$ is a lattice monomorphism of the collection of open subsets of $X$ into the lattice of closed ideals of $C_0(X)$. (It is, in fact, surjective, but the proof of that should wait.)

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $I$ be an ideal in $L^\infty (X, \Omega , \mu )$ that is weak* closed. Show that there is a set $\Delta$ in $\Omega$ such that $I = \left\{ \phi \in L^\infty (X, \Omega , \mu ) : \phi = 0 \text{ a.e. on } \Delta \right\}$.

Let

and let

Show that $\mathcal{A}$ is a Banach algebra and $\mathcal{M}$ is a maximal ideal in $\mathcal{A}$.

Show that for $n \ge 1$, $M_n(\mathbb {C})$ has no nontrivial ideals. How about $M_n(\mathbb {R})$?

Let $\mathcal{A}$ be a Banach algebra but do not assume that $\mathcal{A}$ has an identity. If $I$ is a left ideal of $\mathcal{A}$, say that $I$ is a modular left ideal if there is a $u$ in $\mathcal{A}$ such that $\mathcal{A}(1 - u) \equiv \left\{ a - au : a \in \mathcal{A}\right\} \subseteq I$; call such an element $u$ of $\mathcal{A}$ a right modular unit for $I$. Similarly, define right modular ideals and left modular units. Prove the following.

Using the terminology of Exercise 6, let $I$ be an ideal of $\mathcal{A}$. Show:

If $\mathcal{A}$ is a Banach algebra, a net $\{ e_i\}$ in $\mathcal{A}$ is called an approximate identity for $\mathcal{A}$ if $\sup_i \left\| e_i\right\| < \infty$ and for each $a$ in $\mathcal{A}$, $e_i a \to a$ and $a e_i \to a$. Show that $\mathcal{A}$ has an approximate identity if and only if there is a bounded subset $E$ of $\mathcal{A}$ such that for every $\varepsilon > 0$ and for every $a$ in $\mathcal{A}$ there is an $e$ in $E$ with $\left\| ae - a\right\| + \left\| ea - a\right\| < \varepsilon$.

Show that if $X$ is locally compact, then $C_0(X)$ has an approximate identity.

If $\mathcal{H}$ is a Hilbert space, show that $\mathcal{B}_0(\mathcal{H})$ has an approximate identity.

If $G$ is a locally compact group, show that $L^1(G)$ (1.11) has an approximate identity.

For $0 < r < 1$, let $P_r: \partial \mathbb {D} \to [0, \infty )$ be defined by $P_r(z) = \sum_{n=-\infty }^\infty r^{\left|n\right|} z^n$ (the Poisson kernel). Show that $\{ P_r\}$ is an approximate identity for $L^1(\partial \mathbb {D})$ (under convolution).

If $\mathcal{H}$ is a Hilbert space and $P$ is a projection, show that $\mathcal{B}_0(\mathcal{H})P$ is a closed modular left ideal of $\mathcal{B}_0(\mathcal{H})$. What is the associated right modular unit?

Find the minimal proper left ideals of $M_n(\mathbb {F})$.

Find the minimal closed proper left ideals of $\mathcal{B}_0(\mathcal{H})$, $\mathcal{H}$ a Hilbert space. How about for $\mathcal{B}_0(\mathcal{X})$, $\mathcal{X}$ a Banach space?

What are the maximal modular left ideals of $\mathcal{B}_0(\mathcal{H})$, $\mathcal{H}$ a Hilbert space?