Banach Algebras

Goal: Riesz Functional Calculus.

Definition 8.0.1   A Banach algebras is an algebra (vector space with a multiplication) with a norm such that

1. the normed vector space is a Banach space
2. $\Vert ab \Vert \leq \Vert a\Vert \cdot \Vert b\Vert$

If the Banach algebra has an identity $e$, we assume $\Vert e\Vert = 1$.

Monday, April 10, 2006:

Examples of Banach Algebras:

1. $B(X)$, $X$ a Banach/normed space where multiplication is composition
2. $C(X)$, $X$ a compact Hausdorff space with pointwise multiplication
3. $B_0(X)$, $X$ a Banach/normed space.
4. $C_0(X)$, $X$ is locally compact and Hausdorff.
5. $L^\infty(X,\Omega, \mu)$, $\mu$ a $\sigma$-finite measure.
6. $G$ $\sigma$-compact, locally compact group. Let $m$ be a right Haar measure. That is, it's a nonnegative Borel measure such that $m(U) = m(gU)$ for all $g \in G$ and $m(U) > 0$ for all $U$ nonempty and open. Because the topology is $\sigma$-compact, $m$ is regular. Then, $L^1(G,$    Borel sets $, m)$ is a Banach algebra with
   $$(f \star g)(x) = \int f(x y^{-1}) g(y) dm(y)$$
   This is called $L^1(G)$.

Unitization:
Every Banach algebra can be enlarged to a unital Banach algebra by considering $A \oplus \mathbb{F}$ with $\Vert (a,c)\Vert := \Vert a\Vert + \vert c\vert$. and

$$(a,c) \cdot (b,d) = (ab + ad + cb, cd)$$

Then, $(0,1)$ is an identity for $A \oplus \mathbb{F}$. There are other choices of norms that work.