Banach algebra stuff

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Section 12.1 Banach algebra stuff

Note: developing a discussion of the GNS construction and full spectral theorem.

Here be the Axiom of Choice.

Theorem 12.1.1. Zorn’s Lemma.

If every totally ordered subset of a partially ordered set has an upper bound, then the partially ordered set has a maximal element.

Definition 12.1.2.

A Banach algebra is an algebra \(\A\) (for us over \(\C\)) with a norm \(\norm{\cdot}\) that makes \(\A\) a Banach space and

\begin{equation*} \norm{ab} \leq \norm{a}\norm{b} \end{equation*}

for all \(a,b \in \A\text{.}\)

We’ll assume that our Banach algebras have unit.

The idea of a Banach algebra is to make sure that the multiplication structure is compatible with the norm (i.e. the multiplication should be continuous). Banach algebras are a natural sort of idea - thinking of matrices as representing linear transformations, matrix multiplication (on square matrices) represents composition. It turns out that for a Hilbert space \(\HH\text{,}\) the space \(\B(\HH)\) of bounded linear operators on \(\HH\) is a Banach algebra with the multiplication operation given by composition of maps.

As in classical complex analysis, we can define a notion of analytic functions taking values in a Banach space \(\L\) - a function \(f\) is analytic on a domain \(D \subset \C\) if \(f\) can be expanded as a power series in a neighborhood of every point \(\la \in D\text{.}\)

Theorem 12.1.3.

Let \(A \in \L\text{.}\) Then the resolvent set \(\rho(A)\) is an open subset of \(\C\text{.}\) The resolvent map \(A \mapsto \la I - A\) is an analytic function of \(\la\) on \(\rho(A)\text{.}\)

Proof.

This is a consequence of the fact that the invertible elements are an open subset of a Banach algebra.

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