# Banach lecture 4

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

### Excerpts

For a banach algebra and , if the spectral radius , then is quasi-nilpotent.

Exercise/Results

- If is commutative, then the map is continuous.
- The Volterra algebra is where multiplication is given by , 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 denote the family of equivalence classes of function that are holomorphic on an open set containing . For each HFC gives an element such that

- depends only on the equivalence class of f.
- if is a rational function, then is just the formal evaluation of f at a.
- (spectral mapping thm)
- Given compact such that , then such that where is the norm on the algebra of cont. functions on
- If g is holomophic in an open nbd cont. then

### I.7 Gelfond Theory

Deals with multiplicative linear functionals ; i.e maps that respect the product structure, in the case B is unital these are ring homorphisms.

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- Published:
- September 4, 2009 / 1:02 pm

- Category:
- Banach Algebras (Course)

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