Download e-book for iPad: Banach Algebras and Several Complex Variables by John Wermer (auth.)

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By John Wermer (auth.)

ISBN-10: 1475738781

ISBN-13: 9781475738780

ISBN-10: 1475738803

ISBN-13: 9781475738803

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Extra info for Banach Algebras and Several Complex Variables

Sample text

1. 2. We define two maps from Coo Coo, --+ A l(Q), 0 and a. For E + Bf = df, if f E Coo. We need some notation. Let I be any r-tuple of integers, I = (il' i z , .. , i r ) , 1 ::; i; s n, all j. Put Note. of Thus dz, E N(Q). Let J be any s-tuple (j l ' . . ,is)' 1 ::; ik ::; n, all k, and put dZ J So dZ J E = dz j , 1\ .. 1\ dz j , . N(Q). Then dz, 1\ dZJ E N+S(Q) . For I as above, put III = r. Then IJI = s. 3. Fix integers r, s ~ O. J the sum being extended over all alJ E Coo. I, J with III = r, IJI = s, and with each An element of N,S(Q) is called aform oftype (r, s).

Y = {IJ;,I > 1} is a neighborhood of y. Then 3Y1, . , Yk so that %Yl U ... U % Yk :::> So · Writefj for fyi' Put U = {Ifd < 1,.. ,lfkl < I}. Then U is a neighborhood of x and Un So = 0. Fix a boundary f3 and suppose that f3"'U fails to be a boundary. Then 3f E ff maxxlfl = 1, with maxp"ulfl < 1. Assertion. 3n so that maxxlf'1il < 1, i = 1, ... , k. Grant this for now. Since So is a boundary, we can pick x E So with If(x)1 = 1. By the assertion, IHx)1 < 1, i = 1, .. , k. Hence x E U, denying U n So = 0.

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. Condition (5) alone is not sufficient, as is seen by taking, with D = 21 {zllzl ~ I}, = {f E A(D)If'(O) = O}. Then zZ E 21, z ¢ 21, but (5) holds.

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Banach Algebras and Several Complex Variables by John Wermer (auth.)

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