# Banach Lecture 7

This lecture ( banach7 ) was given on 9/11/09

### Excerpts

Recall for , I get a functional on via . Recalling Thm I.7.1 from lecture 5, if then , meaning is quasi nilpotent. Suppose , then for some . It follows that . Hence cannot be invertible, meaning is not quasinilpotent.

The Gelfond transformation gives a representation of , it has no quasi nilpotents we say is semisimple.

Thm I.7.2. Let be unital, . and holomophic in a nbd of . Then where this is all meant in the sense of HFC (lecture 4)

is a generator of if polynomial in are dense in . It is a rational generator if the functions (HFC) where is a rational function whose poles don’t lie in are dense in .

Prop. I.7.1 Let be unital, a generator then the map of is a homemorphism to .

Prop. I.7.2 Same statement for rational generators.

### I.8 Gelfond Theory Examples

Consider with convolution as the multiplication. Define by . Then is a unit, and is a rational generator. From , conclude the closed unit disk. On the other hand, if , then

since tends to 0, so in fact , the unit circle.

## About this entry

You’re currently reading “Banach Lecture 7,” an entry on Math Meandering

- Published:
- September 13, 2009 / 4:54 pm

- Category:
- Banach Algebras (Course)

- Tags:

## No comments yet

Jump to comment form | comment rss [?] | trackback uri [?]