Cohen-Rudin characterization of homomorphisms of measure algebras

4.2.

COHEN--RUDIN CHARACTERIZATION

OF HOMOMORPHISMS

OF

MEASURE ALGEBRASt

Let

L~T)

be the Lebesgue

on the unit circle

~

MCT)

space and

is a commutative

and the norm of total variation, algebra N of M(T) ~N

and

V~

M~[)

and

L(T)

Banach algebra with the convolution product

is embeded in M ~ )

as a closed ideal.

is said to be an L-subalgebra if it is a closed subalgebra of

A subM~T)

and

, that is, w is absolutely continuous with respect to ~ implies ?.EN .

Let A'(N) be the set of all homomorphisms trivial).

the set of all bounded regular Borel measures

Then, by Shreider

ized character

{~:~ENI

of N to the complex numbers

[I], for every ~ , ~ A C N )

, there corresponds

(which might be a unique general-

or zero system such that

T In the following we shall use the same notation @ for { ~ } . ~:~ ~ A / ~ satisfies, by definition, (i)

~EL~CI~I)and

(ii)

~

= ~v v-a.e,

A generalized character ~ =

jl~-e.ss sl~oJ~?J>O; if v ~ V;

~,v-L,.{s.t).

(iii) ~ , ~ ($+t) =~9(S)~y(~

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.

f'

=~ ~ . ~=

Put ~ = ~

Let

be a Bernoulli convolution product, where ~(a) is a Dirac measure concentrated We fix such a ~ and denote by N(~) the smallest L-subalgebra containing ~. THEOREM to

NC~) 9

([2, 3]).

Suppose

and v in M.

Let M be an L-subalgebra

(A) l~(n) l2 = l~(n) l, i.e.,

L(~)

or N(~) and P be a homomorphism

integer m and a finite subset ~=i~+~,~,,,'",~l

of ~

,

(b) ZeE(M) (j=~,~,...,~), tSATORU IGARI.

2116

Mathematical

Institute,

of M

l~v(n , t) l2 = {~v(n, t) l v-a.e, for all n in

Then we have

(a) a positive

on a point a.

Tohoku University,

Sendai 980, Japan.

(c) ~ e A ( M )

with

l~j l2

=

l~j I (j = I, 2,... ,m) such that

(2) where C E denotes the characteristic function of the set E. Conversely if {~(n)} is a sequence in A'(M) satisfying (A), (a), (b), and (c), then the mapping T given by (I) is a homomorphism of M to M(]') . When M=L[~) then ~(m)={@t~t:~r For this case the'theorem is due to Rudin.

and the condition (A) is obviously satisfied. In the other case, when M = N(~), the theorem is

proved by Igari and Kanjin. Since m~[) is an ideal, our theorem holds good for M = L ( ~ N ( ~ ) We remark that we cannot expect the conditions (a), (b), and (c) without the hypothesis (A) (cf. [2]). Problem I.

For what kind of L-subalgebra M does the above theorem hold good?

Problem 2.

Let

M=M~)

and ~ be a homomorphism of M ~ )

to m ~

.

9

Let {,(n)} be a

sequence of ~'(M~)) given by (I) and assume that {~v(n)} satisfies the condition (A) for a measure ~. Then, characterize v such that the conditions (a), (b), and (c) hold for { ~ ( n ) } . LITERATURE CITED I ~

2. 3.

Yu. A. Shreider, "The structure of maximal ideals in rings of measures with convolutton," Mat. Sb.~ 27, 297-318 (1950). W. Rudin, "The automorphisms and the endomorphisms of the group algebra of the unit circle," Acta Math., 95, 39-56 (1976). S. Igari and Y. Kanjin, "The homomorphisms of the measure algebras on the unit circle" (to appear).

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