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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-l. I<£. II g 11=1 there exists and an Without loss of generality, We now show that some subsequence of {w } is interpolating. Suppose to the contrary, there exists a distinguished homomorphism „eMJG) and iigii = i = Using another i«yg)i.

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() = <£(f) for all e M(G). Of course, if a selector function s exists such tha t f(s(z)) is a (Lebesgue) measurable function for all 44 MILLER, OLIN A N D THOMSON feH°°(D), of then it induces a unital choice, one easily constructs representation many functions However, we want to emphasize the fact desired representation, (Note, by the first one must in the \ s:dD-+M(D) t h a t in order to X- such Using the axiom that s(z) for such a function have f(s(z)) is a measurable function for eM (D).

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A Galois theory example by Brian Osserman

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