Banach Lecture 12

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


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).


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