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.
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