A R K I V F O R M A T E M A T I K B a n d 7 n r 43 1.68 03 1 Communicated 14 February 1968 by L E ~ T CARLESO~
On a strong form of spectral synthesis
BY JOHN E. GILBERT
When G is a locally compact abelian group with character group F, the Fourier Transform of a function / in the group algebra LI(G) is defined b y
f(~)= fJ(x)(x,;~)dx (hEr),
and, for a given closed set ~ in F, I ( ~ ) denotes the closed ideal in LI(G) of all func- tions for which f = 0 on ~ . Following Wik ([8], p. 56) we say t h a t
1. ~ is a Ditkin set i/ to each / E I ( ~ ) there corresponds a sequence { # ~ } c 1(~) where fin =0 in a neighbourhood o / ~ and II/-/~n~/ll--->0 as n--->c~.
2. ~ is a Strong Ditkin set i / t h e sequence {#~} in condition 1 can be chosen inde- pendently o/ / E I ( ~ ) .
I f ~d(F) denotes the discrete coset-ring of F, i.e., the Boolean algebra generated b y cosets of all subgroups of F whether closed or not, we prove in this note:
Theorem 1. Let F be a separable, metrizable group. Then each closed subset ~ o / F in
~d(F) is a Strong Ditkin set.
Using Theorem 1 we obtain immediately the converse of a result of Rosenthal ([6], Theorem 3.1, p. 187) completing the characterization of Strong Ditkin sets in R ~, T ~, ... without interior points.
Theorem 2. Let F = R ~, T ~ or any compact, metrizable group such that the union o / a l l o/ i t s / i n i t e subgroups is everywhere dense. Then a closed set ~ c F having no interior points is a Strong Ditkin set i / a n d only i / ~
eZd(P).
The principal step in the proof of Theorem 1 is the description of closed sets in Zd(F): every closed subset ~ of F (not necessarily separable, metrizable) in ]Ed(F ) is the finite union of sets of the form ~t(II\A)where ~tEF 1, I I is a closed subgroup of F and A belongs to the coset-ring Z ( I I ) of YI (Gilbert [1], Theorem 3.1); conversely, every such union is a closed subset of F in Zd(F). Thus, in R n for example, a closed set ~ without interior points is a Strong Ditkin set precisely when
1 The group operation in all groups is written multiplicatively.
(1)
with F a finite (possibly empty) set and ~ , an affine transformation of a set (I), differing from R ~ • •
•
b y at most finitely m a n y cosets of R ~ (here 0 ~ p = p ( i ) < n , p + q ~<n, a~E R and R ~ is interpreted as the trivial subgroup {0}).The examples given b y Wik a n d l~osenthal follow easily from (1) as do the examples given b y Rosenthal for sets which fail to be Strong Ditkin sets ([6], pp. 187, 8).
First we introduce some notation: for a closed subgroup H of G the norm on the group algebra LI(H) is written ]I(')[[H while t h a t on Li(G) or on the measure algebra M(G)is written [[(.)[[. The H a a r measure on G/H is adjusted so t h a t
for every k in the space :~(G) of continuous functions with compact support in G.
The m a p p i n g / - ~ / ' defined b y
/'(x')= f /(x~)d~ (x'EG/H), 3H
is norm decreasing homomorphism from LI(G) onto LI(G/H) (cf. Reiter [2], p. 415) which can be extended to the respective measure algebras M(G), M(G/H) (Rudin [7], Theorem 2.7.2; Reiter [5]). The Fourier Transform of /' is the restriction of f to the annihilator group of H in F.
Proo/ o/ Theorem 1. I n view of the known structure of closed sets in Za(F) a n d the fact t h a t finite unions of Strong Ditkin sets are again Strong Ditkin sets (Wik [8], Theorem 3) we need only show t h a t a set of the form ]] \A, A E Z(II), is a Strong Ditkin set in F. For then, certainly, a n y translate ~t([[\A) and hence a n y closed set E Za(P) is a Strong Ditkin set. Since already we know t h a t a n y ~ E Za(P) is a Ditkin set (Gilbert [1], Theorem 3.9), given a n y e > 0 and g E I ( ~ ) , there exists g e E I ( ~ ) such t h a t
(a) ~ = 0 i n a n e i g h b o u r h o o d o f ~ , (b) ]lg-g~ii<e. (2) Now let ~ = H \ A , A E E ( I I ) . When H is the annihilator group of II in G, let #n be the idempotent measure in the measure algebra M(G/H) whose Fourier-Stieltjes Transform is the characteristic function of A (as a subset of [I). Further, let # be a n y measure in M(G) for which the restriction to II of the Fourier-Stieltjes Trans- form of/~ coincides with t h a t of/~a. Now assume for the m o m e n t t h a t the following result has been proved.
L e m m a 3. For each A > 1, there is a sequence {~n}c M(G) such that
(i) II nll <A,
(ii) the Fourier-Stieltjes Trans/orm o/ v~ is 1 in some neighbourhood o/H, (iii) ]or any/eLl(G),
lim cSn~n-)e/ldx<ASazHlt'(x')ldx"
(3)572
ARKIV FOR MATEMATIK. B d 7 n r 4 3
Under the conditions of P there is a sequence {an} ~LI(G) forming an a p p r o x i m a t e identity i.e., lla~II = 1 and
lll-an (-/ll o
a s n ~ o o (/EL~(G)). (4) Since, clearly, t e ~ a n E / ( ~ ) , choose a sequence{~n}c
I ( ~ ) for which(a) ~ n = 0 i n a n e i g h b o u r h o o d o f ~ , (b) ][v,~-te-)eanll<l/n. (5) Setting te= =an-(V~-)ea=-v=~%) (n = 1, 2 .... ),
we can soon check t h a t f i ~ = 0 in some neighbourhood of ~ . On the other hand, for a n y [ ELl(G),
lll-a -III + (l-te I)II +AIIIII ll,'n-te a ll,
i.e., b y (3), (4), (5)
lira
lll-te.- lll<A I ll'-te' l'Id/'. (6)
n - + ~ J G I n
B u t # ' = # a and, if / E I ( ~ ) , then tea ~-/' = [', for clearly when IH(g2) is the closed ideal in LI(G/H) of functions whose Fourier Transforms vanish on ~ (as a subset of II),
/' E I~(~), IH(~)=tea~eLI(G/H)
(cf. [4], p. 561). Hence, b y (6), i f / E I ( ~ 2 ) , limn-~]l/-ten~r = 0 which proves t h a t is a Strong Ditkin set.
Remarks. L e m m a 3 is only a mild reworking of a generalization b y Reiter (un- published, but see Rciter [3], L e m m a 2) of a result of Calderon (cf. ]~udin [7], Theorem 2.7.5). Yrom Reiter's version of L e m m a 3 it follows, in particular, t h a t if
~2 c IF[ is a Ditkin set in II then ~ is a Ditkin set in P. Theorem 1 shows, in effect, t h a t a similar result holds with Ditkin set replaced b y Strong Ditkin set at least when ~EF~(II). I n general, however, the result fails for, as Rosenthal points out, a finite interval in R 1 is a Strong Ditkin set in R 1 b u t it cannot be a Strong Ditkin set in R 2 since it t h e n has no interior points and does not belong to the discrete coset- ring Za(R~). Notice that, even so, it is a Ditkin set in R 2. I t is, perhaps, worth noting t h a t the proof of Theorem 1 for an a r b i t r a r y Strong Ditkin set fails because the rate a t which
becomes 0 or negative depends on ] a n d it should be clear t h a t the sequence {te~}
cannot always be chosen independently of /E I ( ~ ) .
Proo/ o/Lemma 3. When {Kn} is an increasing sequence of compact symmetric sets in H for which U n K n = H , let {Un) be a nested sequence of symmetric neigh- bourhoods of the identity e in F/H z shrinking to e and such t h a t ] 1 - (x, y)] < 1/n, x EK=, 7 E U n. Following the construction of a p p r o x i m a t e units in LI(H) as in Reiter ([2], p. 405) we obtain, for each A > 1, a sequence {Tn}cLi(H) satisfying
(i)
f I-~n(~)ld~ <~A,
(ii) i',~ = 1 in a n e i g h b o u r h o o d of e in I ' / I I , (iii) s u p f I T n ( ~ - l ~ ) - v,(~)ld~ <
2A/n.
~e Kn J t l
W h e n each T~ is regarded as a measure on G, the first two of these conditions give t h e corresponding conditions (i), (ii) of the lemma. F o r condition (iii) suppose first t h a t (3) has been established for each k E :/~(G). Then, b y (i), for a n y e > 0 a n d k E ~ ( G ) satisfying
II/-kH <~,
lim
f( ]v=-~/]dx< lim {f l dx+ll/-kll 113=11.}
n-->cr L ~ n--~oo
i.e., (3) t h e n holds for / since s was arbitrary. N o w only minor modifications are required of the second a n d third steps in Reiter's proof of his L e m m a 2 in [3] to obtain (3) for k E ~(G). F o r clearly
Ilk~v=ll.< f. fHk(X )d I
I f, k(x,7)d, + (2A/n) ll kxll,, (7)
for all sufficiently large n since t h e function
k(x~l)
vanishes outside some fixed c o m p a c t set as x ranges over the s u p p o r t of k a n d ~/ over H (in fact, t h e c o m p a c t set C -1- C f3 H where C is t h e s u p p o r t of K). As b o t h sides of (7) are functions onG/H
we m a y integrate b o t h sides with respect todx'
obtaining~ tl kx-~ v. ]].dx= s { f . l f kA,) v.(, l d,} dx
Taking the limit as n ~ ~ we o b t a i n (3). This completes the proof of the lemma.
The University, 1Vewcastle.upon.Tyne, England
574
ARKIV F(iR MATEMATIK. B d 7 n r 4 3
R E F E R E N C E S
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6. ROSENTHAL, H . P., O n t h e e x i s t e n c e of a p p r o x i m a t e i d e n t i t i e s i n ideals of g r o u p a l g e b r a s . A r k i v f6r M a t . 7, 185-191 (1967).
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T r y c k t d e n 26 n o v e m b e r 1968 Uppsala 1968. AImqvist & Wiksells Boktryckeri AB
