Banach lecture 4

This lecture (Banach4) was given on September 2nd 2009.

Excerpts

For $B$ a banach algebra and $a \in B$, if the spectral radius $r(a) = 0$, then $a$ is quasi-nilpotent.

Exercise/Results

1. If $B$ is commutative, then the map $a \mapsto r(a)$ is continuous.
2. The Volterra algebra is $L^1([0,1])$ where multiplication is given by $f*g(t) = \int_0^t f(t-s) g(s) ds$, it is a nonunital commutative Banach algebra.  Prove it is a radical Banach algebra: all its elements are quasi nilpotent.

I.6 Holomorphic Functional Calculus (HFC)

Let $O(sp(a))$ denote the family of equivalence classes of function that are holomorphic on an open set containing $sp(a)$.  For each $f\in O(sp(a))$ HFC gives an element $f(a) \in B$ such that

1. $f(a)$ depends only on the equivalence class of f.
2. if $f(z)$ is a rational function, then $f(a)$ is just the formal evaluation of f at a.
3. $sp(f(a)) = f(sp(a))$ (spectral mapping thm)
4. Given $L \subset \mathbb{C}$ compact such that $sp(a) \subset int (L)$, then $\exists A_L >0$ such that $|f(a)| \le A_l |f|_{C(L)}$ where $|-|_C(L)$ is the norm on the algebra of cont. functions on $L$
5. If g is holomophic in an open nbd cont. $sp(f(a) = b)$ then $g(b) = (g \circ f) (a)$

I.7 Gelfond Theory

Deals with multiplicative linear functionals $B \to \mathbb{C}$; i.e maps that respect the product structure, in the case B is unital these are ring homorphisms.