# Banach Lecture 12

This lecture (banach12) was given on 9/23/09

### Excerpts

$B$ is unital $C^*$-algebra, $a$ is normal element.  $B_a =$ closed subalgebra gen. by polynomial in $a,a^*$; its commutative.

$C(sp(a)) \to B_a$

$h \mapsto h(a)$

where $\widehat{h(a)} = h$ (think about this, it makes sense).  This map is injective, surjective, algebra isomorphism, and preserves $*$ involution.

Extremely disconnected: closure of every open set is open.

Prop I.9.5 $B$ is a unital Banach algebra, $A \subset B$ is closed untial sub alg. $a \in A$.  Then either $sp_A(a) = sp_B(a)$ or $sp_A(a) \backslash sp_B(a)$ is the union of bnd complementary components of $\mathbb{C}\backslash sp_B(a)$.

Prop I.9.6 $B$ is a unital $C^*$ alg. , $A \subset B$ untial $C^*$ alg, $a \in A$.  Then $sp_B(a) = sp_A(a)$.