Characterization of C(x) among its Subalgebras by R. B. Burckel

By R. B. Burckel

Booklet by way of Burckel, R. B.

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U F. = X j=l J and Proof: AIFj = C(Fj) Let X the point adjoined, FO in A A of j. Then A=CO(X). be the one point compactification of xo extended to for each X {xol. as 0 at Let AO x O' and set X, be the functions A = ~ + AO. Then is evidently a uniformly closed, point-separating subalgebra C(X) which contains the constants. C C(FO ) C + AIFj and the cover F j (j = 0,1,2, ••• ) X. 1 ensures that follows easily that j X A = C(X). comprises all the functions in A = Co(X). 12 Let Y be a locally compact Hausdorff space, A uniformly closed, point-separating subalgebra of {Ya ) Let be a family of closed subsets of A\Ya = Co(Ya ) for every a.

With Then g(x) 1= g(y). 11 to learn that is a compact subset of X = Co(X). 13 Let X be a compact Hausdorff space, A closed, point-separating subalgebra of the constants. Suppose that each C(X) = C(F). x ~: Then A x Fx such that = C(X). There are (distinct) points each point which contains x E X, with at most finitely many exceptions, has a compact neighborhood AIFx a uniformly xl, ••• ,xn in the locally compact set Xo = x\{xl, ••• ,xn } has a compact neighborhood of the indicated type. the ideal of functions in Evidently B and I A which vanish on is uniformly closed.

0, we have = Re un there exist Let If As II 1/, Re A is complete in /lun - un .... 0 and /lvn - vII .... O. ~en f,g E B with,say, u = Re f, v = 1m f, u· = Re g, v· = 1m g, we have /lfg/l = lI(uu' - vv' > + i{uv' + u'v>/I = lIuu' - vv'lI + /luv' + u'v/l ~K"u/l I/u'll +Kllv/l/lv'/I +K/lu/l /lv'/I +K/lU'/I /lvll = K(/lull+llv/l> = K/lfll IIgll. Therefore for all real e and taking the supremum on e Next we present a working lemma about analytic functions which we need here and also later on.

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