lf fd/ I f f @l oo} weighted Fourier algebras

Sets of synthesis and sets of interpolation for

weighted Fourier algebras

U. ]~. TEwAI~I

Institut Mittag-Leffler, Djursholm, Sweden

w 1. Introduction. L e t A0(T ) denote the Banach algebra of continuous func.

*ions with absolutely convergent Fourier series. We define An(T ) = { f E C ( T ) : ~

lf(n)t

(1 + [hi) ~ < c~}, a > 0 .

n

We shall also be concerned with the Banach algebra of Lipschitz functions An(T), As(T ) and (A n N A)(T). We let A0(T ) = C(T) and ~I(T) = CI(T), [6;

pp. 42--43].

Let ~ e- C(T) be a regular Banach algebra such t h a t the maximal ideal space of /~ is T. For a closed subset E of T, we define

I ' ( E ) = { f e t ~ : f = O

on E } ,

_R(E) = R/IR(E)

is the restriction algebra of _R to E .

where .R'

f e _d(E),

B(E) = {f C C(E)

: sup

~eM(E)

~ 0

is the dual of R. _R(E)

[]f[I~

is defined b y

ilYlla =

I f f @l oo}

[I/~IIw <

is called the tilda algebra of _R(E). For

lf f d /

sup

u : ~ O

L e t I be a closed ideal in /~, then hull I is defined to be the set of common zeros of all functions in I. We say t h a t a closed subset E of T is of

synthesis

in if

IR(E)

is the only closed ideal in R whose hull is E a n d t h a t

ideal theorem

holds for E

in ~ if every closed ideal I in R whose hull is E is the intersection of all closed primary ideals containing I. We let

206 ARKIV F()~, 3 f . A T E M A T I K . VO]. 9 :No. 2

PM~(T) -~ (A~(T))'

and M~(T) = (A~(T))'

Let us remind ourselves t h a t when we talk of A~, the index ~ E [0, oo[ while in ease of 2~ a n d 2 ~ N A , a E [ 0 , 1 ] .

An element of P M ~ ( T ) will be called an ~-pseudomeasure and an element of M~(T) will be called an a-measure. B y PM~(E) [M~(E)] we shall denote the set of a-pseudomeasures [c~-measures] carried by E. E is called a set without true a-pseudomeasures if PM~(E) = M~(E) and E is called an Ha-set if A~(E) = A~(E).

I n Section 2, we shall discuss the Ditkin property for the algebras A~ and 2~ Cl A and prove the following:

THEOREM 1. Let E be a closed subset of T such that for an infinite sequence N ~O

{ j}}=l qf integers, the points 27~j/N (N E {Nj}) either belong to E or are at least at a distance 2 s I N from E. Then E is a set of synthesis for 2 ~ n A , 0 < o ~ < 1.

I n Section 3, we describe a totally disconnected perfect set of synthesis for A~(T), 0 < ~ < 1/2.

Finally in Section 4, we discuss H~-sets and sets without true ~-psendomeasures.

We prove t h a t only finite sets are Ha-sets. I n this process we shall also prove t h a t every function f E A~(E) can be extended to A~(T) without increasing its norm and t h a t the tilda algebra of A~(E) is A~(E).

I wish to express m y deep gratitude to Professor N. Th. Varopoulos for his generous advice a n d criticism and to the I n s t i t u t Mittag-Leffler for its hospitality.

w 2. Carl Herz [1] has proved t h a t if f E A~ is such t h a t f(to) = if(to) . . . fM(t0) = 0 t h e n there exists a sequence of functions f , E A~ such t h a t f , = 0

in a neighbourhood of t o and

I l f - - f f . i b = ~ 0 a s n - ~ o o .

We state the following consequence of his result.

TItEOEE~. Let E and F be closed subsets of T such that E is of synthesis in A~ and F has countable boundary. Then ideal theorem holds for E U F in A~.

I n particular, ideal theorem holds for a closed subset with countable boundary for the algebra A~.

I f 0 < c~ < 1, we observe t h a t usual ])itkin property holds for the algebra X~ Cl A. We shall now give the proof of Theorem 1.

For a positive integer n and f E 2~ Cl A, let us define f , C 2~ Cl A as follows:

f,(2:~j/n) = f(2z~j/n), 0 < j < n and linear in each interval [2~j/n,

2~(j + 1)In].

S E T S O F S Y : N T t I E S I S A I ~ D S E T S O:F I ~ T E I ! , P O L A T I O N : F O R V e ' E I G t t T E D : F O U I ~ I E I ~ A : L ( ~ E B R A S 207

L e t T , : ~ f3 A - + 2~ 13 A be t h e linear operator defined b y T,df) = fn. The r o u t i n e c o m p u t a t i o n shows t h a t I]T.][ < 3 for e v e r y n. Since piecewise linear functions are dense in X~ [3 A it follows t h a t i f iv C a n infinite sequence {Nj}

of integers, T N f - - > f in 2 ~ [ 3 A for all f E 2 ~ [ 3 A as i V - - > ~ .

L e t E a n d {Nj} be as in t h e s t a t e m e n t of Theorem 1. The condition on E implies t h a t if iV e{iVj} t h e n T N f = 0 on E for all f eI;~md(E). More- over, T N f vanishes on an open set which contains all b u t finitely m a n y points of E.

Given s > 0, choose iv E {iVj} so large t h a t l l f - <

N o w t h e D i t k i n p r o p e r t y for ~ [3 A implies t h e existence of a g E X~ 13 A such that g. T N f vanishes in a n e i g h b o u r h o o d of E a n d

IiT f - g " TNflb < .

Therefore I I f - g" TNfll < e where gTz~f = 0 in a n e i g h b o u r h o o d of E. This proves t h a t E is of synthesis for t h e algebra 2~ [3 A.

w 3. W e shall follow McGehee [4] in constructing a perfect t o t a l l y disconnected set of synthesis for t h e algebra A~, 0 < a < 1/2.

W e recall t h e following l e m m a due to McGehee [4] a b o u t f i n i t e l y s u p p o r t e d measures.

3.1. LE~MA. Let F = { x j : 1 < j <_k} be a finite set of distinct points of T.

Then given s > 0 there exists a number N N ( x 1 . . . x k ; s) such that for every

e M ( F )

m a x

I~(n)l

>_ (1 - s)II~llP~, for each m;

In--ml << N

where

II~tlPM = sup T/~(n) I

rt

W e construct E as t h e intersection of a decreasing sequence of closed sets E k, each E k being t h e u n i o n of s~ disjoint closed intervals each of length lk. L e t Ek = {x (1) , 9 9 9 x (sk)} be t h e set of the left end points o f the intervals c o n s t i t u t i n g E k. Given Ek, choose N k (by L e m m a 3.1) such t h a t for e v e r y m e a s u r e # E M(Ek),

sup L (n)l > 89 I~l -< Nk

Iqow choose t h e l e n g t h lk of t h e intervals in E k such t h a t t h e points of E~ are a t least 21k a p a r t a n d such t h a t

= o(1) (1)

W e shall n o w show t h a t E is a set of synthesis for t h e algebra A~, 0 G ~ < 1/2.

208 ~ Z K ] Z V F~JZ~ 1VIATE1VI2~TIK. VO1. 9 ^l'q'o. 2 3.2. L ~ A . To each S E PM~(E)

gk E M(Ek) such that

I S ( n ) - ~ ( n ) i

<C(~)lnI(skl~)~/'iiSli~

for all 1~ and n where C(~) is a constant depending only on ~.

I n particular,

we can associate a sequence of measures

(2)

lim ~k(n) ---- S(n) for all n . (3)

/~ --> co

Proof. We observe t h a t the formal integral of S is the L~-function S(n) e i"~ (where we have assumed t h a t S(0) 0) with the norm

~(x) ~ Z - - =

n r s n

[Jail2 _< C(c~)[ISl[p~ where C(~) is a positive constant depending only on a. The proof of the lemma is now exactly the same as t h a t of the lemma 4, p. 141--143 in [4].

3.3. TEEO~E~. E is a set of synthesis for the algebra A~, 0 _< ~ < 1/2.

Proof. Let S e PM~(E).

for all n

Therefore

B y (2) in L e m m a 3.2, we have IS(n) ^-

~(u) l _< c(~)Inl(sA)x/2ilSilpM~

and k. N o w b y our choice

sup I~k(n) l > 89 [I/~IIPM

19k(n) t

lf[~k[IPMc~ < 2 SUp < 211 "3 U C(O~)Nk(Sk~k) 112] liSlIPMo~

-- i~1<~, k (1 + In]) ~ ^--

~ o w since Nk(sklk)l/2~-0(1), it follows t h a t sup [l~tk][eM~ < ~ . This together k

with (3) of L e m m a 3.2 implies t h a t #k converges to S in the weak*-topology of PM~. This proves that E is of synthesis for the algebra A~.

Remark. B y changing the thinness condition (1), we can construct sets E which are of synthesis for the algebras A~, where ~ E [n, n -J- 1/2[, n is a positive integer.

w 4. We start b y stating the following proposition which relates the sets without true a-pseudomeasures and the Ha-sets.

4.1. :PRO:POSITIOn. Let E be a closed subset of T without true o~-pseudomeasures.

Then E is an Ha-set and the ideal theorem holds for E in A~.

S E T S O F SYI~TIIESIS A I ~ D S E T S O F I N T E R F O L A T I O I ~ FOI% W E I G H T E D ~ O U I % I E R A L G E B I % A S 209

W e n o w proceed to determine the H~-sets. W e need the following easy l e m m a . 4.2. LEM~-~. Let or > 0 and

l~ = sequences c = { c ~ } : s u p ~ ( 1 + lnl) ~ < oo , l~ = {sequences a = {a~}: ~ [a.l(1 § In[) ~ < oo},

n

c0,~ = {c e 12 : c~ = o ( I n l ~ ) } .

Then (Co,~)' = 11.

The following proposition is of i n d e p e n d e n t interest. H o w e v e r , we shall use t h e m e t h o d of its proof.

4.3. PROPOSITION. Let E be a closed subset of T. Then for each f 6 A s ( E ) there exists F 6A~(T) such that f [ E - - - - f and [[F[[A~(T)= ItfNAAE ).

Proof. L e t N~(E) b e the annihilator of I~(T)(E) in PM~(T) and L ( E ) = Co, ~ fl N~(E). T h e n f induces a b o u n d e d linear functional f ' on L(E) with the n o r m ~ NfI[A~(E)" B y the H a h n - B a n a c h t h e o r e m there is an extension F ' of f ' to Co, ~ such t h a t ]IF'I[ ~ llflI~(E). [By L e m m a 4.2 this corresponds t o a function

2' C A~(T) such t h a t [[F[[Aa(T ) ~_~ lIf[iAa(E)" W e shall n o w show t h a t FIE = f. L e t t o b e an a r b i t r a r y point of E. Then O,o 6 L(E). Therefore

f(to) = f'(~,.) = F(~,0) = F(t0).

This completes t h e p r o o f of t h e proposition.

Remark. The idea of the a b o v e p r o o f goes b a c k to Y. K a t z n e l s o n a n d K.

D e L e e u w [3].

W e n o w p r o v e the m a i n proposition n e e d e d for our s t u d y of H~-sets. The proposition is interesting for its own sake.

4.4. PROI'OSITIOZ~. Let E be a closed subset of T. Then A s ( E ) = A~(E).

Proof. L e t f 6 A~(E). T h e n f induces a b o u n d e d linear functional f ' on M(E) as a subspace of N~(E). As in P r o p o s i t i o n 4.3 we can e x t e n d f ' to Co. ~ a n d t h u s get F 6 A,(T) such t h a t FIE ~ f, i.e. f 6 A s ( E ). This completes the proof.

W e n e e d the following easy facts: (i) L e t E be a closed subse~ o f T. Then for 0 < ~ < 1, 2~(E) = As(E ) .

(ii) I f E is an infinite c o m p a c t subset o f T, t h e n 2~(E) 4 : A s ( E ) for 0 < ~ 1 .

210 AI~KIV FOR MATE~IATIK. VO1. 9 N o . 2

T h e a b o v e facts t o g e t h e r w i t h P r o p o s i t i o n 4.4 i m p l y

4.5. PROPOSITION. I f E is an infinite compact subset of T then A~(E) =~ ~ ( E ) . O u r findings a b o u t H~-sets c a n be s u m m e d u p in t h e following:

4.6. T~EORE~. Let E be a compact subset of following are equivalent:

(i) E is a set without true ~-pseudomeasures.

(ii) E is an H~-set.

(iii) E is finite.

T and 0 < ~ < 1. T h e n the

References

1. HE,z, C. S., Two problems in the spectral synthesis of unbounded functions. Mimeographed notes (1964).

2. KArAtE, J. P., & SAI.E~, R., Ensembles Tarfaits et s~ries trigonometriques, Hermann, Paris, 1963.

3. I)ELEEuW, K., & KATZNELSO~, Y., On certain homomorphisms of quotients of group algebras.

Israel J. Math. 2 (1964), 120--126.

4. MCG~HEF~, 0. C., Certain isomorphisms between quotients of a group algebra. Pacific J.

Math. 21 (1967), 133--151.

5. VA~OrOULOS, N. T~., On a problem of A. Beurling. J. Functional Analysis 2 (1968), 24--30.

6. ZYG~U~D, A., Trigonometric series, 2nd ed., Cambridge University Press, 1959.

Received May, 1970 U. B. Tewari

Indian Institute of Technology Kanpur (U.P.)

India