Banach lecture 10

This lecture ( ) was given on 9/18/09.

Excerpts

Some basic results were given on Pontryagin duality.  Then it was:

I.9 C^* Algebras

An involution of a Banach algebra $B$ is a map $a \mapsto a^*$ such that

1. $(ca)^* = \bar c a^*$
2. $(a + b)^* = a^* + b^*$
3. $(ab)^* = b^*a^*$
4. $(a^*)^* = a$

Some examples were given.  A $C^*$-algebra is a Banach algebra equipped with an involution such that

$||a^*a|| = ||a||^2$

Results; in a $C^*$ algebra

1. $1^* = 1$
2. $||a|| = ||a^*||$
3. The left and right regular representations are isometric.
4. A unit can be adjoined to any nonunital $C^*$-algebra to produce a unital $C^*$-algebra.