# Banach Lecture 6

the lecture Banach6

### Excerpts

First came a review of wk-* topologies on the dual of a Banach space. For example

Alaoglu’s Thm: Let be a Banach space and give the wk-* topology. The closed unit ball in is compact.

It follows that all closed balls in are wk-* compact and that all wk-* closed and bounded sets are compact.

For a unital Banach algebra, the Gelfond topology on unit ball in is the induced topology from the wk-* topology. Using the thm above it can be shown that is a compt. Hausdorff space. with its Gelfond topology is called the Gelfond space of or the maximal ideal space or the structure space or the spectrum.

In the nonunital case, construct Gelfond space of and idenitify with where is the functional with as its kernel.

The Gelfond transformation is a map which sends $lates a \to \hat a$ where send .

## About this entry

You’re currently reading “Banach Lecture 6,” an entry on Math Meandering

- Published:
- September 9, 2009 / 10:21 am

- Category:
- Banach Algebras (Course)

- Tags:

## No comments yet

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