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Igari S.-Cohen-Rudin characterization of homomorphisms of measure algebras

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188宝金博页面版:  cohen-rudin characterization of homomorphisms of measure algebras：（cohen-rudin表征测量代数的同态）

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内容提示： 4.2. COHEN--RUDIN CHARACTERIZATION OF HOMOMORPHISMS OF MEASURE ALGEBRASt Let L~T) be the Lebesgue space and M~[) the set of all bounded regular Borel measures on the unit circle ~ MCT) is a commutative Banach algebra with the convolution product and the norm of total variation, and L(T) is embeded in M~) as a closed ideal. A sub- algebra N of M(T) is said to be an L-subalgebra if it is a closed subalgebra of M~T) and ~N and V~ , that is, w is absolutely continuous with respect to ~ implies ?.EN . Let A'(N)...

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4.2. COHEN--RUDIN CHARACTERIZATION OF HOMOMORPHISMS OF MEASURE ALGEBRASt Let L~T) be the Lebesgue space and M~[) the set of all bounded regular Borel measures on the unit circle ~ MCT) is a commutative Banach algebra with the convolution product and the norm of total variation, and L(T) is embeded in M~) as a closed ideal. A sub- algebra N of M(T) is said to be an L-subalgebra if it is a closed subalgebra of M~T) and ~N and V~ , that is, w is absolutely continuous with respect to ~ implies ?.EN . Let A'(N) be the set of all homomorphisms of N to the complex numbers (which might be trivial). Then, by Shreider [I], for every ~ ,~ACN) , there corresponds a unique general- ized character {~:~ENI or zero system such that T In the following we shall use the same notation @ for {~}. A generalized character ~ = ~:~ ~A/~ satisfies, by definition, (i) ~EL~CI~I) and jl~-e.ss sl~o J~?J>O; (ii) ~ = ~v v-a.e, if v ~ V; (iii) ~,~ ($+t) =~9(S)~y(~ ~,v-L,.{s.t). A Let ~ be a homomorphism of N to M[T) Then the mapping ~--~(~V){~), YEN, defines a homomorphism for every integer n, where "^" denotes the Fourier--Stieltjes transform T Thus there exists a generalized character ~(~=[%,C~,~):y~N 1 or zero system such that A C v) T Let {an}n~ 0 be a sequence of integers such that a n > 2 and an > 2 for infinitely many n. Put ~=~ ~= f' =~ ~. Let be a Bernoulli convolution product, where ~(a) is a Dirac measure concentrated on a point a. We fix such a ~ and denote by N(~) the smallest L-subalgebra containing ~. THEOREM ([2, 3]). Let M be an L-subalgebra L(~) or N(~) and P be a homomorphism of M to NC~) ?9 Suppose (A) l~(n) l 2 = l~(n) l, i.e., l~v(n , t) l 2 = {~v(n, t) l v-a.e, for all n in and v in M. Then we have (a) a positive integer m and a finite subset ~=i~+~,~,,,'",~l of ~ , (b) ZeE(M) (j=~,~,...,~), tSATORU IGARI. Mathematical Institute, Tohoku University, Sendai 980, Japan. 2116

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