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Banach algebra on which A Proof. [t(A), put continuous integer k and original to c h e c k one and that that if equivalent is a p - n o r m from A onto to the o r i g i n a l A° and with let I N " I[A P one. be P3 =0 • F i x define N a IIA= p( II a II~, k II Pa [I) It is e a s y there A ° = { a 6 A : a(x o) = 0} projection and p6 P semisimple II " IIA k for a 6 A. is a p - n o r m is s u f f i c i e n t l y on large A, equivalent then N" I[A to the is 48 submultiplicative. 3. Banach Proposition.

ChA: be a net of e l e m e n t s of X converging ~ ( x i) = s i £ C h A , xi=- 6s. + A U i ' var(A~i) -< e for i 6 I U {0}. 1 Without loss of g e n e r a l i t y s I £ 8A and (Aui)i61 the measures tends + A~ ° 6s we can so A = and since and so6 ChA that in t h e w e a k 6s * (si)i61 topology tends to to AU 1. H e n c e and + A~ I r e p r e s e n t the same functionals I v a r ( A u I) < l i m v a r ( A u i) ~ e < I so w e g e t show the o on assume sI• b) ~ is a c l o s e d To prove Let b) we (xi)i61 xl'~ ~ s.

Ii) + (e) ~ Let us II X • f " g - where <__ II X 6 A -I t (iv) for Banach al~ebras. 11ell < ~(I+~) 2 hence f,g in A (22) and (23) II 11 - ell < e. For all 30 Illl - x-lll <_ ~ + E ( I + ~) 2 = ~" and consequently <_E'(1- . f-g-f×gEl f,g Step 8. Proof. in Since for r e a l any real form to p r o v e the b e an S 5E such that there measure for Let S p = 6s Ap + Ap s o on s o~ identified Hausdorff with space S an it is proposition. be a c o m p a c t functional c a n be compact of the a l g e b r a is a m e a s u r e the algebras.