Examples of Banach Algebras

Let

$${\ell }^{\mathrm{\infty }}(\mathrm{\Omega }):=\{f:\mathrm{\Omega }\to \mathbb{C}:\Vert f{\Vert }_{\mathrm{\infty }}<\mathrm{\infty }\},$$

where $\mathrm{\Omega }$ is a set and

$$\Vert f{\Vert }_{\mathrm{\infty }}:=\underset{\omega \in \mathrm{\Omega }}{sup}|f(\omega )|.$$

Claim: ${\ell }^{\mathrm{\infty }}(\mathrm{\Omega })$ is a unital Banach algebra.

 

Proof: It is clear that that ${\ell }^{\mathrm{\infty }}(\mathrm{\Omega })$ is unital and an algebra. Let $(f_n{)}_n$ be Cauchy, let $ϵ>0$, let $N$ be such that

$$\Vert f_n-f_m{\Vert }_{\mathrm{\infty }}=\underset{\omega \in \mathrm{\Omega }}{sup}|f_n(\omega )-f_m(\omega )|<\frac{ϵ}{2}$$

for all $n,m\ge N$, define $f:\mathrm{\Omega }\to \mathbb{C}$ by $\omega \mapsto \underset{n}{lim}{f}_n(\omega )$, and note that $f$ is well-defined since $(f_n(\omega ){)}_n$ is Cauchy for all $\omega \in \mathrm{\Omega }$. Then

$$\underset{m}{lim}|f_n(\omega )-f_m(\omega )|=|f_n(\omega )-\underset{m}{lim}{f}_m(\omega )|=|f_n(\omega )-f(\omega )|\le \frac{ϵ}{2}<ϵ$$

for all $\omega \in \mathrm{\Omega }$, meaning that $\Vert f_n-f{\Vert }_{\mathrm{\infty }}<ϵ$ for all $n\ge N$. Finally,

$$\begin{array}{}◼& \Vert f{\Vert }_{\mathrm{\infty }}=\Vert f-f_n+f_n{\Vert }_{\mathrm{\infty }}\le \Vert f_n-f{\Vert }_{\mathrm{\infty }}+\Vert f_n{\Vert }_{\mathrm{\infty }}<\mathrm{\infty }.\end{array}$$

Let

$$C_b(\mathrm{\Omega }):=\{f\in {\ell }^{\mathrm{\infty }}(\mathrm{\Omega }):f\text{ is continuous}\},$$

where $\mathrm{\Omega }$ is a topological space.

Claim: $C_b(\mathrm{\Omega })$ is a unital Banach algebra.

Proof: We show that it is closed: recall that convergence with respect to $\Vert \cdot {\Vert }_{\mathrm{\infty }}$ is equivalent to uniform convergence and that uniformly convergent sequences of continuous functions converge to continuous functions.

$$\begin{array}{}◼& \end{array}$$

Let

$$C_0(\mathrm{\Omega }):=\{f\in C_b(\mathrm{\Omega }):f\text{ vanishes at infinity}\},$$

where $\mathrm{\Omega }$ is locally compact and Hausdorff.

Recall that $f$ vanishes at infinity if for all $ϵ>0$, there is $K$ compact such that $|f(\omega )|<ϵ$ for all $\omega \in K^c$.

Claim: $C_0(\mathrm{\Omega })$ is a Banach algebra.

Proof: We show that it is closed: let $n$ be such that $\Vert f_n-f{\Vert }_{\mathrm{\infty }}<ϵ/2$ and let $K$ compact be such that $|f_n(\omega )|<ϵ/2$ for all $\omega \in K^c$. Then

$$|f(\omega )|\le |f_n(\omega )-f(\omega )|+|f_n(\omega )|<\Vert f_n-f{\Vert }_{\mathrm{\infty }}+\frac{ϵ}{2}<ϵ$$

for all $\omega \in K^c$.

$$\begin{array}{}◼& \end{array}$$

$C_0(\mathrm{\Omega })$ is one of the most important examples of a Banach algebra in C*-algebra theory, and it is unital if and only if $\mathrm{\Omega }$ is compact, in which case

$$C(\mathrm{\Omega })=C_b(\mathrm{\Omega })=C_0(\mathrm{\Omega }).$$

Let $L^{\mathrm{\infty }}(\mathrm{\Omega },\mu )$ be the set of classes of essentially bounded, complex-valued, measurable functions on $\mathrm{\Omega }$, where $(\mathrm{\Omega },\mu )$ is a measure space, equipped with the essential supremum norm.

Claim: $L^{\mathrm{\infty }}(\mathrm{\Omega },\mu )$ is a unital Banach algebra.

Let $B_{\mathrm{\infty }}(\mathrm{\Omega }):=\{f\in {\ell }^{\mathrm{\infty }}(\mathrm{\Omega }):f\text{ is measurable}\}$, where $\mathrm{\Omega }$ is measurable.

Claim: $B_{\mathrm{\infty }}(\mathrm{\Omega })$ is a unital Banach algebra.

Proof: Recall that a point-wise convergent sequence of measurable functions converges to a measurable function.

$$\begin{array}{}◼& \end{array}$$

Let $A$ be the set of continuous, complex-valued functions on the closed unit disc $D\subset \mathbb{C}$ that are holomorphic on the interior $D^{\mathrm{o}}$ of $D$.

Claim: $A$ is a unital Banach algebra called the disc algebra.

Proof. If $(f_n{)}_n$ converges to $f$ with respect to $\Vert \cdot {\Vert }_{\mathrm{\infty }}$, then it converges uniformly and $f$ is thus continuous. Moreover,

$$0=\underset{n}{lim}{\oint }_{\gamma }{f}_n(z)\text{d}z={\oint }_{\gamma }\underset{n}{lim}{f}_n(z)\text{d}z={\oint }_{\gamma }f(z)\text{d}z$$

for all closed, piece-wise $C^1$ curves $\gamma$ on $D^{\mathrm{o}}$. Therefore, by Morera's theorem, $f$ is holomorphic on $D^{\mathrm{o}}$.

$$\begin{array}{}◼& \end{array}$$