By John Wermer (auth.)
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Additional resources for Banach Algebras and Several Complex Variables
E. S. Transl. 1 (1955). The proof given here is due to L. Waelbroeck, Le Calcul symbolique dans les algebres commutatives, J . Math. Pure Appl. 33 (1954), 147-186. 2 for the general case was proved by R. Arens and A. Calderon, Analytic functions of several Banach algebra elements, Ann. Math. 62 (1955), 204-216. 4 is a special case of a more general result given by Arens and Calderon, loe. cit. 6 and its corollaries are due to Silov, loc. cit. 4 has followed Hormander's book [40, Chap. 3J. 2 see Waelbroeck, loe.
Pr+ 1 in en and consider ¢ E ~p,q(Q), o some neighborhood of pn(pl> ... ,Pr+ 1)' We first sketch the argument. Step J. Embed pn(pl' , Pr+ t> in r-: 1(P1> . . , Prj by the map u:z -. (Z, Pr+ I (zj). p, are polynomials in ZI' ... , Zn+ 1 which do not involve Zn+ I ' Let L denote the image of pn(pl ' .. , Pr+ t> under u. n denotes the projection (z, Zn+ I) -. Z from e n+ 1 -. en. Note n o u = identity. Step 2. Find a a-closed form <1>1 defined in a neighborhood of r-: I(PI' . , Pr) with <1>1 = ¢(n) on L .
X n ) . AI,. We throw together all the corresponding Yj and call them C l, . . , Cm , and we let B be the closed subalgebra generated by Xl" '" x,; C l, . AI" and so 3u l, . . , Un E B such that Lj UiXj - oej ) is invertible in B. Hence a ¢ O"B(Xl> " " x n). Thus O"B(Xl> " " x n) c lv, proving the assertion. 2 holds in general. As a first application we consider this problem. Let 21 be a Banach algebra and X Em:. , when can we find Y E 21 with y Z = x? An obvious necessary condition is the purely topological one: (5) 3y E C(A) with y Z = x on A.