By R. B. Burckel
Publication by way of Burckel, R. B.
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Extra info for Characterization of C(x) among its Subalgebras
12 (Bernard ) Let X be a compact Hausdorff space, A and B Banach algebras lying in B is conjugate closed and Proof: and If so does is u1traseparating, 1 E A C B, then B = C(X). II II X s: II II A liB' so it is easy to apply the Closed Graph Theorem to see that the inclusion of Therefore A As noted before (III, p. 1) we have "x s: II II C(X). A C B. B. Since A A into B is continuous. separates the points of IN x X, Moreover an application of the (real) Closed Graph Theorem in B, using as before conjugation is continuous in is conjugate closed: B.
5 If u E Re A Proof: ~ If u > 6 and By compactness u on ~ X then 5 > 0 on logou E Re A. X 5 > O. for some is any non-zero complex homomorphism of ~ A, then is a continuous functional of norm 1 (see III, p. 1) and so lifts to a functional of norm 1 on Theorem. C(X) by the Hahn-Banach This extension is represented by a (complex, regular, Borel) measure U on X so is a positive measure. I< 1) is such that and Re f 5 > O. W is Now if 9 ~ W(g) a non-zero complex homomorphism of 9 ~ and the subalgebra With t 1 ~ f ~l(f) Re A i~ ~2(f) > 0 right half plane.
We set (9) ~ = 1m f(l) - A > 0, and then choose a partial sum of f about o in p in the power series expansion such that in the compact subset we have (10) Ip(z) - f(z) I < min(~,l-~) Then ( 11) and for all p(O) = f(O) o by (6) and (3) 35 (12) (10) (7) IRe p(z) I ~ IRe f(z) 1+lp(z)-f(z) I < IRe f(z) 1+1-11 s 11+(1-11)=1 (13) 1m p(l) ~ 1m f(l)-IIm p(l)-Im f(l) I ~ 1m f(l)-Ip(l)-f(l) I (10) (9) > 1m f(l) - 6 = A. 5 If u E Re A Proof: ~ If u > 6 and By compactness u on ~ X then 5 > 0 on logou E Re A.