Abelian Banach Algebras

Let $\mathcal{A}$ be a Banach algebra with identity and let $J$ be the smallest closed two-sided ideal of $\mathcal{A}$ containing $\{ xy - yx : x, y \in \mathcal{A}\}$; $J$ is called the commutator ideal of $\mathcal{A}$.

Using the terminology of Exercises 2.6 and 2.7, let $\mathcal{A}$ be an abelian Banach algebra without identity and show that if $M$ is a maximal modular ideal, then there is a homomorphism $h: \mathcal{A} \to \mathbb {C}$ such that $M = \ker h$. Conversely, if $h: \mathcal{A} \to \mathbb {C}$ is a nonzero homomorphism, then $\ker h$ is a maximal modular ideal. Moreover, the correspondence $h \to \ker h$ is a bijection between homomorphisms and maximal modular ideals.

If $\mathcal{A}$ is an abelian Banach algebra and $h: \mathcal{A} \to \mathbb {C}$ is a homomorphism, then $h$ is continuous and $\left\| h\right\| \le 1$. If $\mathcal{A}$ has an approximate identity $\{ e_i\}$ such that $\left\| e_i\right\| \le 1$ for all $i$, then $\left\| h\right\| = 1$ (see Exercise 2.8).

Let $\mathcal{A}$ be an abelian Banach algebra and let $\Sigma$ be the set of nonzero homomorphisms of $\mathcal{A} \to \mathbb {C}$. Show that $\Sigma$ is locally compact if it has the relative weak* topology from $\mathcal{A}^*$ (Exercise 3).

With the notation of Exercise 4, assume that $\mathcal{A}$ has no identity and let $\mathcal{A}_1$ be the algebra obtained by adjoining an identity. For $a$ in $\mathcal{A}$, let $\sigma (a)$ be the spectrum of $a$ as an element of $\mathcal{A}_1$ and show that $\sigma (a) = \{ h(a) : h \in \Sigma \} \cup \{ 0\}$. Also, show that the maximal ideal space of $\mathcal{A}_1$, $\Sigma_1$, is the one-point compactification of $\Sigma$.

With the notation of Exercise 4, for each $a$ in $\mathcal{A}$ define $\hat{a}: \Sigma \to \mathbb {C}$ by $\hat{a}(h) = h(a)$. Show that $\hat{a} \in C_0(\Sigma )$ and the map $a \mapsto \hat{a}$ of $\mathcal{A}$ into $C_0(\Sigma )$ is a contractive homomorphism with kernel $= \bigcap \{ M : M \text{ is a maximal modular ideal of } \mathcal{A}\}$.

If $X$ is locally compact, show that $x \mapsto \delta_x$ is a homeomorphism of $X$ onto the maximal ideal space of $C_0(X)$.

Let $X$ be locally compact and for each open subset $U$ of $X$ let $C_0(U) = \left\{ f \in C_0(X) : f(x) = 0 \text{ for } x \text{ in } X \setminus U\right\}$. Show that $C_0(U)$ is a closed ideal of $C_0(X)$ and that every closed ideal of $C_0(X)$ has this form. Moreover, the map $U \mapsto C_0(U)$ is a lattice isomorphism from the lattice of open subsets of $X$ onto the lattice of ideals of $C_0(X)$.

With the notation of the preceding exercise, show that $C_0(U)$ is a modular ideal if and only if $X \setminus U$ is compact.

If $\mathcal{A}$ is an abelian Banach algebra and $a \in \mathcal{A}$, say that $a$ is a rational generator of $\mathcal{A}$ if $\{ f(a) : f \text{ is a rational function with poles off } \sigma (a)\}$ is dense in $\mathcal{A}$. Show that if $a$ is a rational generator of $\mathcal{A}$, then $\Sigma$ is homeomorphic to $\sigma (a)$.

Verify the statements made in the following example. If $V: L^2(0,1) \to L^2(0,1)$ is the Volterra operator and $\mathcal{A}$ is the closure in $\mathcal{B}(L^2(0,1))$ of $\{ p(V) : p \text{ is a polynomial in } z\}$, then $\mathcal{A}$ is an abelian Banach algebra and $\text{rad}\, \mathcal{A} = \text{cl}\{ p(V) : p \text{ is a polynomial in } z \text{ and } p(0) = 0\}$. In other words, $\mathcal{A}$ has a unique maximal ideal, $\text{rad}\, \mathcal{A}$.

Say that $a_1, \ldots , a_n$ are generators of $\mathcal{A}$ if $\mathcal{A}$ is the smallest Banach algebra with identity that contains $\{ a_1, \ldots , a_n\}$. Show that $a_1, \ldots , a_n$ are generators of $\mathcal{A}$ if and only if $\mathcal{A} = \text{cl}\{ p(a_1, \ldots , a_n) : p \text{ is a polynomial in } n \text{ complex variables } z_1, \ldots , z_n\}$, and if $\Sigma$ is the maximal ideal space, then there is a homeomorphism $\tau$ of $\Sigma$ onto a compact subset $K$ of $\mathbb {C}^n$ such that if $p$ is a polynomial in $n$ variables, then $\gamma (p(a_1, \ldots , a_n)) = \tau^\# (p)$.

Verify the statements made in the following example. Let $\mathcal{A}$ be the closure in $C(\partial \mathbb {D})$ of the polynomials in $z$. If $\Sigma$ is the maximal ideal space of $\mathcal{A}$, then $\Sigma$ is homeomorphic to $\sigma_{\mathcal{A}}(z)$ (here $z$ is the function whose value at $\lambda$ in $\partial \mathbb {D}$ is $\lambda$). Now $\sigma_{\mathcal{A}}(z) = \text{cl}\, \mathbb {D}$. If $f \in \mathcal{A}$, then the Maximum Modulus Theorem shows that $f$ has a continuous extension to $\text{cl}\, \mathbb {D}$ that is analytic in $\mathbb {D}$; also denote this extension by $f$. The continuous homomorphisms on $\mathcal{A}$ are of the form $f \mapsto f(\lambda )$ for some $\lambda$ in $\text{cl}\, \mathbb {D}$.

(Zelazko) Let $\mathcal{A}$ be an algebra and suppose $\phi : \mathcal{A} \to \mathbb {C}$ is a linear functional such that $\phi (a^2) = \phi (a)^2$ for all $a$ in $\mathcal{A}$. Show that $\phi$ is a homomorphism.

Let $\mathcal{A}$ be an abelian Banach algebra with identity that is semisimple [that is, $\text{rad}\, \mathcal{A} = (0)$]. If $\left\| \cdot \right\|$ is the norm on $\mathcal{A}$ and $\left\| \cdot \right\|_1$ is another norm on $\mathcal{A}$ that also makes $\mathcal{A}$ into a Banach algebra, then these two norms are equivalent.

Let $\mathcal{A}$ be the closure in $C(\partial \mathbb {D})$ of the polynomials in $z$ and let $K = \left\{ \phi \in \mathcal{A}^* : \phi (1) = \left\| \phi \right\| = 1\right\}$. Show that $\text{ext}\, K = \left\{ \delta_z : \left|z\right| = 1\right\}$.

Show that $f(x) = \exp (\pi i x)$ is a generator of $C([0,1])$ but $g(x) = \exp (2\pi i x)$ is not.

Show that $C(\partial \mathbb {D})$ does not have a single generator though it does have a single rational generator (that is, an element $a$ such that $\{ r(a) : r \text{ is a rational function with poles off } \sigma (a)\}$ is dense).