# Banach Lecture 11

The lecture: banach11

### Excerpts

Unless otherwise specified, $B$ is a unital $C^*$-alg.

Prop. I.9.1 If $a$ is a normal element (commutes with adjoint) in a $C^*$ alg. then $||a|| = r(a)$.

Prop I.9.2 If $a \in B$ is self adjoint, then $sp(a) \subset \mathbb{R}$.

Prop I.9.3 If $B$ is commutative then $\widehat{a^*} = \overline{\hat a}$.

Thm I.9.1 If $B$ is commuative, then $\widehat{B} = C(\mathscr{M}(B))$.  The Gelfond transform is an isometry and it intertwines the involution with complex conjugation.

Cor. $B$ is nonunital, commutative then $\mathscr{M}(B)$ is loc. compact and $\widehat{B} = C_0(\mathscr{M}(B))$.

Prop. I.9.4 Let $B$ be commutative and assume polynomial in $a,a^*$ are dense in $B$.  The $\phi \mapsto \phi(a)$ gives homemorphism $\mathscr{M}(B) \to sp(a)$ and $\widehat{B} \cong C(sp(a))$.