By Brian Osserman

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**Additional info for A Galois theory example**

**Sample text**

Fix X € dG and €>0. From the 36 MILLER, OLIN A N D THOMSON proof of Lemma |g(w)|<£ for 32, we all see there exists we(C \A(X,e/2))nG. a function Since Gn

It now follows from the definition of k t h a t 0 = ,(A(f))p(h)-l. Hence, 1 . *(f)p(h); therefore, 7r(f) is invertible and the proof is completed.! Remark 26. containing cr(S). Suppose S is a pure subnormal operator and K is a compact set If G = int K and conditions (ii) and (iii) hold in Theorem 24, then the proof of t h a t theorem exhibits a unital representation (that is unique via Theorem 10) n defined on H°°(G). Remark S = 7r(x) is P u r e 27. an Suppose d 7r is a unital representation defined on H (G) such that SUBNORMAL OPERATORS (28) 31 cr(S)nG is a dominating set for H°°(G).

Mz(G) - {4>eM(G): Hence, *(*)-•}. With this notation we can s t a t e this later characterization as follows: Theorem 7r (x) = X- 42. )-f(s(>)) almost everywhere m. , i(

### A Galois theory example by Brian Osserman

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