By T.V. Tonev

Taken care of during this quantity are chosen issues in analytic &Ggr;-almost-periodic features and their representations as &Ggr;-analytic features within the big-plane; n -tuple Shilov barriers of functionality areas, minimum norm precept for vector-valued capabilities and their functions within the learn of vector-valued features and n -tuple polynomial and rational hulls. functions to the matter of lifestyles of n -dimensional complicated analytic buildings, analytic &Ggr;-almost-periodic buildings and buildings of &Ggr;-analytic big-manifolds respectively in commutative Banach algebra spectra also are discussed.

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**Example text**

9. PROPOSITION. ,. ,f n E B. Then either there exists a 'p E s p B with cp(fj) = 0, j = 1,.. ,n,or there exist n elements gl, . . ,g n in B with C fjgj = e. ~nparticular f E B-' j= 1 if and only if cp(j)# O for every cp E s p B . Chapter I . Let J = { fjg, : gj E B, j = 1,. . , n } be the j=1 ideal generated by f 1 , . . ,fn. e. p(fj) = 0 for all fj. e. if J = B , then e E J and hence e = c f j g j for j= 1 some gj E B , j = 1 , . . ,n, which completes the proof. 10. COROLLARY. Let X be a compact Hausdodspace and B c C ( X ) .

Let J1 be a proper ideal of B which contains Ker cp. We claim that J1 = Kercp. e. the conjugacy classes with respect to the equivalency relation ” a b if and only if a - b E Ker cp”. The coset [a]containing a fixed element a E B is defined by [a]= { b E B : b-a E Ker c p } . The operations 22 ChaDter I. Uniform Alvebras [a]+ [b]= [a+ b], [a][b]= [ab] and A[a]= [Aa] are well defined on f3; provided with them, 0 becomes a commutative algebra over C. There arises a natural projection T : B + 23 defined by ~ ( a=) [a)= { a + b : b E Kery} = a + Kery.

Consequently J1 = ~ - ' ( 3 1 ) . 3 1 is an ideal in B. Indeed, for each c E J1 and a E B we have [c)[a]= [ca]= ~ ( c aE) 3 1 since ca E J1. In addition [el 4 3 1 since if we assume on the contrary that [el E 3 1 = K(J~), then [el = ~ ( c for ) some c E J1. e. c - e E Kercp wherefrom e E J1+ Ker y = J1, which is absurd because, as we saw above, J1 nB-' = 0. We conclude that 3 1 is a proper ideal in the field 23 2 C . Consequently Jl = [O] since if there exists a c E J1 with [c] # [ O ] , then [c] will be an invertible element in B which is contained in the proper ideal 31,which is absurd.