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)).

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