A Class of Commutative Banach Algebras l. Main assumptions and defildtions

HARMONIC ANALYSIS BASED ON CERTAIN BANACH ALGEBRAS

BY

Y N G V E D O M A R in Uppsala

COMMUTATIVE

I n t r o d u c t i o n

C o n t e n t s

Page

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Chapter I . A class o/ commutative Banach algebras. 1 . 1 . F M a i n a s s u m p t i o n s a n d d e f i n i t i o n s . . . 4

1.2. S o m e l e m m a s c o n c e r n i n g t h e s u b c l a s s e s F 0 a n d F ' . . . . 6

1.3. L i n e a r f u n c t i o n a l s o n F . . . 9

1.4. C o m p l e x - v M u e d h o m o m o r p h i s m s o f F . . . 10

1.5. T h e s p a c e of r e g u l a r m a x i m a l i d e a l s . . . 14

Chapter I I . Special algebras and special elements. 2.1. V a r i o u s e x a m p l e s o f B a n a c h a l g e b r a s F . . . 17

2.2. P r o o f o f t h e o r e m 2.11 . . . 20

2.3. T h e c l a s s 9 . . . 25

Chapter I I I . The spaces A and the spectrum. 3.1. T h e s p a c e s A . . . 28

3.2. D e f i n i t i o n a n d m a i n p r o p e r t i e s of t h e s p e c t r u m . . . 31

3.3. T h e o r e m s o n i t e r a t e d t r a n s f o r m a t i o n s . . . 35

3.4. E l e m e n t s w i t h o n e - p o i n t s p e c t r u m . . . 40

Chapter I V . A n equivalent de]inition o] the spectrum. 4.1. E l e m e n t s i n A w i t h a p p r o x i m a t e i d e n t i t i e s . T h e s u b s p a c e A 1 . . . 47

4.2. S o m e l e m m a s . . . 51

4.3. T h e s p e c t r a l s e t s A'a a n d A"a 9 . . . 55

4.4. T h e n a r r o w t o p o l o g y . . . 62

B i b l i o g r a p h y . . . 66

1 - 563802. Acta mathematica. 96. Imprim6 le 2 mai 1956.

2 YNGVE DOMAR

I n t r o d u c t i o n

I n various papers (ef. the bibliography) A. Beurling has studied the harmonic analysis of functions on the real line. Using different approaches he has introduced the notion of the spectrum of a function as a set on the dual real line which, roughly speaking, consists of the frequencies of the characters of which the function can be regarded as composed,

The development of the theory of Banach algebras has made it possible to ex- tend a wide sector of harmonic analysis into more abstract theories. I t has therefore been natural to study spectral theory from a more general point of view. Thus Gode- ment [8] gave a definition of the spectrum valid for bounded measurable functions on a Ioeally compact Abelian group, and his approach was pursued by, among others, Kaplansky [10] and Helson [9].

Many problems in this field remain unsolved. I t is b y no means obvious to what extent the speetral theory depends on the metrical properties of the real line and on the structure of the function spaces which were considered by Beurling. The reason is perhaps that Beurling attained his results by means of a very large variety of methods.

Algebraic arguments are sometimes used (as in [1]), but more often methods from the theory of analytic functions and potential theory are applied, and not all these methods are available in the general setting. Especially the generalization to groups has met m a n y obstacles. More progress has been made when the theory has been restricted to the real line (Warmer [17]).

Beurling has given several more or less equivalent definitions of spectrum. The one which most easily lends itself to generalizations is the definition in [5], which defines the spectrum of a function in L ~ as the set of frequencies of the characters which are included in the weak closure of the linear manifold spanned by the transla- tions of the function. This definition was also used by Godement. I n the theory which can be developed from this definition, it is of fundamental importance t h a t L 1 be a eommutative Banaeh algebra under convolution with the dual group as the regular maximal ideal space and with the property t h a t every proper closed ideal is ineluded in at least one regular maximal ideal. The lasli-mentioned property is closely connected with the general Tauberian theorem (Wiener [18]) as is shown e.g. in Loomis [11] w167 25D, 37A.

A closer study of the possibilities offered by the above definition reveals t h a t the concept of transformation is of a fundamental importance in the development of the theory. I t is essential t h a t L 1 can be considered as an algebra of transformations of

H A R M O N I C A : N A L Y S I S B A S E D ON C E I ~ T A I N C O M M U T A T I V E B A N A C I t A L G E B R A S 3 L ~176 into itself, if the transformation is defined as the ordinary convolution. I n fact, even the definition of the spectrum can be expressed in terms of these transforma- tions, and this opens the way to generalization in the following direction.

Let A be a normed linear space and F a commutative algebra with a representa- tion onto an algebra of linear transformations of A into itself. For every a E A and ]E F we denote by [ o a the corresponding transformed element in A. We assume t h a t F is normed in such a way t h a t

Ill o all<- IIslI" I1, 11

and we suppose t h a t F is a Banach algebra under this norm, with a space S of regular maximal ideals. We define, for every a E A, the spectrum Aa as the subset of S consisting of all regular maximal ideals which contain the closed ideal of all ] E F for which ] o a = O. (0 denotes the null element in A.) We assume that every proper closed ideal is included in at least one regular maximal ideal, and then an e m p t y Aa implies that i o a = 0 for every ] E F. Let us finally assume t h a t this is true only if a = 0. Then Aa is e m p t y only if a = 0, and this fundamental uniqueness theorem gives us a solid basis for a general theory.

Of course very few of the problems in the Beurling spectral theory can be formulated in this abstract setting. The notion of translation has for instance dis- appeared as a main ingredient of the definition. The lost connections with the Fourier analysis can, however, partially be recovered if we assume t h a t S is a locally com- pact Abelian group, and further specializations will of course lead us.still closer to the field of study in the Beurling papers.

Our object in this pzper is to study a class of algebras F of the above type, where S is a locally compact Abelian group, and then discuss the corresponding spectral definition. I t will turn out that m a n y of the essential results in the Beurling spectral theory can be approached in our rather general setting, e.g. the characteriza- tion of elements with one-point spectrum (originally studied b y Beurling [2] and later b y Kaplansky [10], Helson [9], Riss [15], Wermer [17] and others) and the spectral definition by means of the narrow closure (Beurling [2]).

As for the methods employed, we naturally have to utilize in a very essential way the general theory of commutative Banach algebras together with the special properties of the class of algebras which we discuss. The elementary Fourier analysis on groups is rather freely used, and results from the theory of analytic and quasi- analytic functions are applied at certain places where it has been possible to restrict the discussion to the real line. I n paragraph 2.2 the structure theory of locally compact

4 YNGVE DOMAR

Abelian groups is used in the discussion of a particular example of the algebras F, but apart from this, the theory does not depend on structure theory.

The first two chapters deal exclusively with the properties of the algebras F . Chapter I I I contains the definition of the spectrum and an account of certain of the most available spectral properties. In chapter IV the discussion centers around an- other definition of the spectrum. I t is proved to be equivalent to the original defini- tion and closely connected with the Beurling definition in [2].

Most of the results in chapter I I I are valid for more general classes of algebras F. The results in chapter IV, however, depend on the structural properties of F, and it is doubtful whether it is possible to prove similar results in greater generality.

I t is assumed t h a t the reader has a certain knowledge of the theory of com- mutative Banach algebras as in Gelfand [7] and parts of the theory of Fourier analysis on locally compact Abelian groups as in Pontrjagin [14], Weft [16] and Godement [8]. Whenever possible, however, we take the liberty of referring to the exposition in Loomis [11], and certain more or less standard arguments in harmonic analysis, such as convolutions, inversion theorems, etc., are used without reference.

Functions on the dual groups G and G are denoted b y / ( x ) , g(x) . . . and /(2), (3) . . . respectively. The only exceptions are the characters (x, ~), where the above notation is inconvenient. Whenever two functions, such as /(x) and ](~), are men- tioned in the same context, they indicate a pair of functions which in some sense are Fourier transforms. Addition is chosen as group operation.

C H A P T E R I

A Class of Commutative Banach Algebras l. Main assumptions and defildtions

Let G be an Abelian locally compact group with the dual group G. I t will be convenient for our purposes to assume t h a t the groups are Hausdorff spaces. This is no essential restriction as is shown in L. 28 D. (L. denotes here and in the following references to the corresponding paragraph in Loomis [11].)

We introduce a Banach space F of complex-valued functions / ( x ) , defined and finite everywhere on G. The addition of two elements in F is defined as the ordinary addition of the two functions and the multiplication of an element with a complex

t t A R M O N I C A N A L Y S I S B A S E D O N C E R T A I N C O M M U T A T I V E B A N A C H A L G E B R A S 5

constant is the ordinary multiplication of the function with the same constant. Dif- ferent functions are supposed to be different elements, and for t h a t reason /(x) is the null-element if and only if [(x)~O.

F u r t h e r m o r e we suppose t h a t if two functions /l(x) and ]2(x) belong to F, then the same is true for the function

1 (x) =/, (x), 12 (x),

and the corresponding norms fulfill the relation

II/11 ~< II 1111" II/2 II

This implies t h a t F is a c o m m u t a t i v e Banach algebra.

For a n y function / ( x ) C F we denote b y A~ the set in G where / ( x ) * O , and b y Ar the closure of A~.

We shall introduce some further assumptions and notations:

I. Suppose that /or every neighborhood N o/ the identity in G there exists a non- negative, not identically vanishing /unction [N (X) in F with the /ollowing properties:

A. AfN

B. IN(X)= S (x,~)fN(2)d2,

where IN(2) is continuous and e LI(G).

C. All continuous /unctions ~ (2) such that

1~(2) l--<lfN(2)l

have the property that the /unctions

g (x) = [. (x, 2) ~ (2) d2 belong to F, and their norms are uni/ormly bounded.

(1.11)

(1.12)

Before we can proceed with our assumptions we h a v e to discuss a consequence of Assumption I.

Using the Pontrjagin duality theorem and the definition of the topologies of the dual groups we see (L. 34C) t h a t for every compact set C c ~ , the set of points x C G, such t h a t for every 2 C C

[ 1 - ( x , 2 ) [ < L

6 Y N G V E D O M A R

is an open set in G. Since it contains the i d e n t i t y o of G, it is a n e i g h b o r h o o d N of o. The function /N(X) is n o n - n e g a t i v e a n d therefore, if 3 E r

If,,,(3) l = I .r l,,,(x) (x, 3) d x I > 89 .r 1,,, (x) dx.

G G

(We assume here a n d in t h e following t h a t the H a a r measures on G and G are n o r m e d in such a w a y t h a t the c o n s t a n t in t h e Fourier inversion formula has t h e value 1.) This shows t h a t it is possible to find, for e v e r y c o m p a c t set C, a f u n c t i o n / N ( x ) such t h a t [/g(3) l has a positive lower b o u n d on C. Using A s s u m p t i o n I C we see t h a t this implies t h a t the class F o o/ /unctions

g (x) = S (x, 3) ~ (3) d2,

where ~ (2) is continuous and vanishes outside a compact set, is a subclass o / F . T h e f o r m u l a /1 (x)./2 (x) = j" (x, 2) d 2 S/1 (2 - 20) ~2 (20) d30,

G

which is t r u e if ~1(2) a n d ~2(2) belong to L 1(6), shows t h a t F 0 is m o r e o v e r a sub- algebra (L. 2 8 A 4).

Our second a s s u m p t i o n will b e : I I . Snppose that F o is dense in F.

W e shall i n t r o d u c e a n o t h e r subclass of F. L e t us first form t h e class of all functions g(x) of t h e t y p e (1.12), for which 9(2) is continuous a n d satisfies (1.11) for some N, a n d for which Ag is compact. Then we denote by F' the class o/ /unc- tions o/ the type ~.g(x), where ~ is an arbitrary constant.

2. Some l e m m a s concerning the subclasses F 0 and F'

The classes F 0 and F ' will p l a y i m p o r t a n t r61es in t h e discussion of the B a n a c h algebra F. F o r later use we shall collect in this section some lemmas on these sub- classes.

I f a function g (x) belongs to F 0 or to F ' , t h e n we shall use t h e t e r m Fourier t r a n s f o r m of g(x) for the continuous function ~(3), which in t h e sense of (1.12) is associated to g(x).

L E M M A 1.21. Consider /or a given compact set C in G the subclass o/ all /unc- tions g(x) EFo, /or which the Fourier trans/orms ~(3) vanish outside ~. Then there exists a /inite constant ds, such that /or all these /unctions

ItARMO~TIC A : N A L u B A S E D O~T C E R T A I N C O M M U T A T I V E B A N A C H A L G E B R A S 7

II g (x) ll <~ d~ ll ~ (3) ll~,

where

lid(3)I[

The proof follows at once f r o m the discussion in 1.1.

L E M M A 1.22. For every neighborhood N o/ the identity 6 in G there exists a /unction /(x) E F' such that the Fourier trans/orm [(3) satis/ies

^ < ' I

o<_/(x)_l

[(6)

1, /1"

[ (3) <_ 89 outside N. ]

P R O O F . L e t us start f r o m a function /N(X) with c o m p a c t N.

negative, and for t h a t reason

/N(X) is non-

The function

is t h e Fourier t r a n s f o r m of a f u n c t i o n E F ' . I t satisfies t h e first t w o of t h e condi- tions (1.21), a n d f u r t h e r m o r e we k n o w t h a t

(x) _ 89

outside a certain c o m p a c t set C, since t h e Fourier t r a n s f o r m of a f u n c t i o n E L 1 (G) vanishes at infinity.

N o w let N be t h e given neighborhood. W e m a y of course assume t h a t it is open. T h e set C 0 of all points in C, which are n o t contained in N, is t h e n a com- p a c t set, n o t containing 6.

L e t us for e v e r y point x E G denote b y 0x t h e open set in G where I 1 + ( x , 3 ) [ 2 < 2 .

I f 3 * 6 there exists a point x 0 such t h a t (x 0, 3)~= 1. A n e l e m e n t a r y reasoning shows t h a t for a suitable value of t h e integer n t h e n u m b e r

(x 0, 3) n = (n Xo, 3) has to satisfy t h e i n e q u a l i t y

I I + ( n x 0 , 2 ) l_< 1.

Therefore t h e sets 0x cover all points in G with t h e exception of 6, a n d as a result we m a y select a finite sub-sequence {0~.v} ~ which covers t h e c o m p a c t set C o.

8 u D O M A R

L e t us n o w f o r m t h e function

1 n

~(pc): ~ ~ I1+(x~, pc)l ~

I t satisfies t h e conditions (1.21), the third one, however, only inside C 0. B u t t h e function

[(Pc) =)i (PC). ~ (PC)

satisfies (1.21) in all details, a n d since ~(2) is a linear c o m b i n a t i o n of characters,

/(x)-- S (x, PC) I(PC)dpc

has c o m p a c t AI, a n d therefore it belongs to F ' .

L ~ t M A 1.23. For every neighborhood N o/ 6 and /or every ~>0 there exists a ]unction g (x) E F' which has the representation

where

g (x) = gl (x) + S (x, PC) ~ (PC) dpc,

G

]lg,[l<~,

and where g2 (PC) is non-negative and continuous, vanishes outside N and satis/ies g2 (PC) d 2 = 1.

N

P R O O F . Let us s t a r t f r o m a function ](x) in F ' which satisfies the conditions in L e m m a 1.22 with respect to ~r. L e t $(PC) be a continuous function satisfying 0 ~< Is (PC)_< 1 a n d which vanishes outside 37 a n d assumes t h e value 1 on t h e set where

t (PC) -> ~.

Choose for e v e r y positive integer n t h e c o n s t a n t dn such t h a t

A p p a r e n t l y

d~ J"/c(k) ]/(PC)In dpc = 1.

lim d 1 / ~ - 1. n - - n--~oo

This relation and A s s u m p t i o n I C h a v e as consequence t h a t the n o r m of t h e function

dn f ( x , ~ ) ( 1 k ( ~ ) ) [ / ( ~ ) ] n d 2

H A R M O N I C A N A L Y S I S B A S E D O N C E R T A I N C O M M U T A T I V E B A N A C I { A L G E B R A S 9

t e n d s t o 0, w h e n n - - - > ~ . L e t us a s s u m e t h a t t h e n o r m is s m a l l e r t h a n ~ for n = n 0.

T h e n t h e l e m m a follows b y choosing

9 ~ ,,k rt a ~,

g~(x) =d~o j" (x, x)(1 - fc(~)) If(x)] dx,

5

~2 (2) = tin. k ( 2 ) [ / ( x ) ] n~

L E M M A 1.24. ]~or every pair o[ sets C and 0 in G, where C is compact, 0 is open and C ~ O, there exists a /unction /(x) E F' such that

o_</(z)<_ 1 in O, / (x) = 1 in C, ] (x) = 0 o u t s i d e O.

P R O OF. W e d e n o t e for e v e r y p a i r of sets E 1 a n d E 2 in G b y E I + E 2 t h e set of all p o i n t s x = x l + x ~ , where x l E E l a n d x ~ . C E 2. T h e n t h e r e e x i s t s (L. 5 F , L. 2 8 A 3 ) a c o m p a c t s y m m e t r i c n e i g h b o r h o o d N of t h e i d e n t i t y in G such t h a t

C + N + N c O .

L e t us a s s u m e t h a t t h e n o n - n e g a t i v e f u n c t i o n /N(X) satisfies t h e r e l a t i o n

G

I f t h i s is n o t t h e case, we m a y c h a n g e t h e f u n c t i o n b y m u l t i p l y i n g i t w i t h a s u i t a b l e c o n s t a n t . T h e n l e t [1 (x) be t h e c h a r a c t e r i s t i c f u n c t i o n of t h e set C ,~ N. The f u n c t i o n

belongs to F ' , a n d i t is v e r y e a s y to v e r i f y t h a t it h a s t h e r e q u i r e d p r o p e r t i e s . 3. Linear functionais o n F

W e a r e going to show t h a t we h a v e a c e r t a i n r e p r e s e n t a t i o n of t h e l i n e a r func- t i o n a l s on F as B o r e l m e a s u r e s on G. H e r e t h e t e r m Borel m e a s u r e is used in t h e wide sense, i.e. i t i n c l u d e s also c o m p l e x s e t - f u n c t i o n s .

S u p p o s e t h a t ]*{1) is a l i n e a r f u n c t i o n a l on F . I f we consider t h e f u n c t i o n s g(x) E F 0 for which t h e F o u r i e r t r a n s f o r m s ~(2) v a n i s h o u t s i d e a f i x e d c o m p a c t set (~, we g e t f r o m L e m m a 1.21

10 YNGV]~ DOMAR

If*<g)l<_ IIf*ll. Ilgll-<de II/*11- II ~(~)11=,

which shows t h a t the functional is at t h e same time a linear functional on the class of functions ~(5) u n d e r t h e uniform norm. Therefore

1" (g) = ~ ~j (~) d/i~ ( - 5),

c

where /J5 is a Borel measure, u n i q u e l y defined on the interior of the c o m p a c t set of points 5 such t h a t - 5 6 C . Since C m a y be chosen arbitrarily we can e x t e n d this result to the following l e m m a :

L E M M A 1.31. To each linear functional /* on F there corresponds a unique Borel measure ~ on G such that if / 6 F o

1"(1)= ~ /(5) d ~ ( - 5 ) .

F u r t h e r m o r e we h a v e the following lemma, which is an i m m e d i a t e consequence of A s s u m p t i o n I I .

L EMMA 1.32. Two different /unctionals can not correspond to the same measure.

N o w let / (x) 6 F ' a n d let /* be a linear functional with the corresponding measure ft. I f we let g (x) r u n t h r o u g h all t h e elements in F 0 such t h a t the Fourier t r a n s f o r m

(5) satisfies

1~(5) l-<1f(5)1,

we get

.r 1l(5) t I d,,; (- 5) 1 = sup I J" 0(5) d/,( -5) 1= sup I/* (g) l -< II f* I1" sup Ilgll < oo

8 8

since the n o r m s of the functions g (x) are u n i f o r m l y b o u n d e d (Assumption I C). T h u s we get

L EMMA 1.33. I / /(X)E F' and if ff is a measure which corresponds to a linear functional on F, then

J I/(5)11dD(-5)l < co.

4. Complex-valued homomorphisms of F

A h o m o m o r p h i s m of F onto the complex n u m b e r s is a m a p p i n g

f (x)+~ (/),

where ~ (/) for e v e r y / (x) 6 F is a finite complex n u m b e r with the following properties :

HARMONIC A ~ A L Y S I S BASED ON CERTAIN COMMUTATIVE BANACH ALGEBRAS l l

i . B.

(c~/1 + c~ 1~) = cl i (/1) + c~ X (1~).

(11" 1~) = ~ (11)" ~ (/~)

for a n y t w o c o n s t a n t s Q and c 2 a n d for a n y two elements ]l(x) a n d /2(x).

C. ~ ( / ) * 0

for a t least some [ (x).

Since F is a c o m m u t a t i v e algebra, Jl([) is bounded, considered as a functional on F (L. 23A). Therefore it is a linear functional on F , a n d in order to d e t e r m i n e t h e c o m p l e x - v a l u e d h o m o m o r p h i s m s of F , we h a v e only to find t h e n o t identically vanishing linear functionals which satisfy B.

Suppose t h a t /* is such a functional a n d suppose t h a t it corresponds to t h e m e a s u r e /~ in t h e sense of L e m m a 1.31.

Since f* is n o t identically v a n i s h i n g a n d because of A s s u m p t i o n I I , t h e r e exists a function /0(x)C F o such t h a t

/* (/0) = f f0 (~) d/~ ( - .~) = 1.

P u t

/o (~o + ~) d ~ ( - ~o) = ~ (~), (1.41) which is a continuous function, satisfying

~ ( ~ ) = l .

(1.42)

L e t /1 (X) be a v a r i a b l e function in F o. T h e relation /* (fo h) = 1" (/o)"/* (fl) = 1" (11) gives t h e f o r m u l a

[f ]1 (~)/o (#o - ~) d #] d/~ ( - #0) = ] ]1 (#) d # ( - ~), a n d using (1.41) this m a y be w r i t t e n

]1 (~) ~ ( - ~) d ~ = ~/1 (~) d ~ ( - ~).

Since this is t r u e for e v e r y /1 E

Fo,

(1.43)

12 Y~GVE DOMAR

As a consequence we have, if /1 (x) and /2(x) are quite a r b i t r a r y functions in Fo, f If fl (2) /2 (20 -- 2) dx] a ( - 20) d 2 o = f fl (2) a ( - x) d x . y f2 (2) ~ ( - 2) dx.

G G ~

And from this relation it is quite easy to see t h a t :i (21 + 22) = ~ (21)" ~ (22) for every pair of points in G. Thus we h a v e :

(1.44)

after Theorem 2.31.) LEMMA 1.42.

such that

L EMMA 1.41. A measure ~, corresponding to a linear /unctional which gives a complex-valued homomorphism, has to satisfy (1.43), where the continuous /unction ~(2) satisfies (1.42) and (1.44).

We shall now proceed to prove the following more precise s t a t e m e n t :

T H E O R E M 1.41. The only /unctionals which give complex-valued homomorphisms, are the /unctionals ] * ( / ) = / ( x ) /or any x E G.

These functionals certainly give complex-valued homomorphisms. The only problem is to verify the condition C, which is, however, an immediate consequence of L e m m a 1.24. We m a y mention t h a t as a consequence these functionals are linear. They correspond in the sense of (1.43) to bounded functions ~(2), i.e. to the ordinary characters (x, 2).

Because of L e m m a 1.32 no other linear functionals correspond to bounded func- tions ~ (2). For t h a t reason the only thing we have to prove is t h a t no linear func- tional corresponds to a measure (1.43), where the continuous function ~(2) satisfies (1.44) and is unbounded. (Concerning the existence of such ~(2), see the r e m a r k

For the proof we need the following l e m m a :

Let 2 o be a fixed point in G and let c be a fixed real number 0 < c < m

Form the open set Oc (20) of all points x E G such that

-c<arg(x, 20)<c

Suppose moreover that f(2) is the Fourier-Stieltjes transform o/ a bounded Borel measure ~, vanishing outside Oc (2o) i~e.

f (2) = f (x, 2) d # (x).

Oc(~.)

H A R M O N I C A N A L Y S I S B A S E D O N C E R T A I N C O M M U T A T I V E B A N A C H A L G E B R A S 13 Then /or every integer n

f ( n ~ 0 ) = f (x,~o)nd#(x) = f e~~

Oc (~o) - c

where b(O) is o/ bounded variation on ( - c , e).

P R O O F 0 F L E M M a 1.42. L e t us d e n o t e b y g (e i 0) a n a r b i t r a r y c o n t i n u o u s func- t i o n on t h e u n i t circle. Consider t h e s p a c e of all t h e s e f u n c t i o n s u n d e r t h e u n i f o r m n o r m . T h e n

~ ( g ) = f g((X,~o))d#(x)

O c (,~o)

is a l i n e a r f u n c t i o n a l a n d t h e r e f o r e i t h a s t h e f o r m k ( g ) = ~ g ( e * ~

where b(O) is of b o u n d e d v a r i a t i o n . B y v a r y i n g g(e i~ i t is e a s i l y seen t h a t b(O) is c o n s t a n t o u t s i d e ( - e , c). A n d t h e n t h e l e m m a follows b y choosing g(e i~ = e in~

I ~ R O O F OF T H E O R E M 1.41. L e t us a s s u m e t h a t a c e r t a i n u n b o u n d e d f u n c t i o n

&(k) of t h e t y p e d e s c r i b e d in L e m m a 1.41 c o r r e s p o n d s to a l i n e a r f u n c t i o n a l on F . W e shall p r o v e t h a t t h i s l e a d s t o a c o n t r a d i c t i o n .

L e t So be a p o i n t such t h a t

I a ( 0)1

T h e n we h a v e for e v e r y i n t e g e r n

] ~ (n 30) [ = d ~.

Choose a n a r b i t r a r y n u m b e r c such t h a t 0 ~ c ~ y t .

T h e set 0c (30), d e f i n e d in L e m m a 1.42, is a n o p e n n e i g h b o r h o o d of t h e i d e n t i t y in G.

T h e r e f o r e we can f i n d a n o t i d e n t i c a l l y v a n i s h i n g f u n c t i o n ] l ( x ) i n F ' such t h a t Af~cOr On a c c o u n t of L e m m a 1.33 we h a v e

L e t us now choose a f u n c t i o n Jo (x)E F o such t h a t

=

fo

G

14 Y N G V E D O M A R

satisfies

f(~)

]. (i.45)

The relation

II(~)I -< f ll~(,~-~o)l.la(-~+~-o)l-la(_~)l.la(~o)

I llo(~,o)l<Z~o

1

"J" 111(~)I la(-~)Id~'sup la(-~o)l.llo(~o)l (1.4o)

-<I~(-~)I e

shows t h a t for some finite constant K

It(~)1-<1~(_~) I

for every ~. I n particular we have for every integer n

^ < K

II(-nxol--d~ ^(1.47)

B u t [(~) is the Fourier transform of the function /l (x) . /o (X), and this function vanishes outside Oc(~o). Therefore we m a y apply L e m m a 1.42, and we then get for every integer n

c

[(n~o) = ~ e - ~ ~ db(O), (1.48)

- c

where b(O) is a function of bounded variation.

(1.47) and (1.48) show t h a t the analytic function

H(rei~ = ~

is regular in the region 1 < [ z [ < d and satisfies lim H (z) ~ 0

r - - ~ l + 0

uniformly in a n y closed interval outside the interval [01_<c. Then it~ has to vanish identically which is contradictory to (1.45). This proves tile theorem.

5. The space o f regular m a x i m a l ideals

L~MMA 1.51. The topology on G is the weakest topology in which all the/unctions / (x) E F are continuous.

H A R M O N I C A N A L Y S I S B A S E D O N C E R T A I N C O M M U T A T I V E BANACI-I A L G E B R A S 1 5

PROOF. L e m m a 1.24 implies t h a t the functions can not be continuous in a n y weaker topology. Thus it only remains to show t h a t every / ( x ) E F is continuous in the topology on G.

All the functionals /(x) have a norm <_ 1, since this p r o p e r t y is always true for a functional t h a t gives a complex-valued homomorphism (L. 23A). Therefore we have

II/(x) II <- II [ II. (1.51)

However, because of Assumption I I we can a p p r o x i m a t e every ](x)E F arbitrarily closely in the F - n o r m b y means of functions in F 0. And thus (1.51) implies t h a t every [ (x)E F can be a p p r o x i m a t e d arbitrarily closely in the uniform n o r m by means of continuous functions, and this has the consequence t h a t every / (x) E F is continuous.

I n the theory of c o m m u t a t i v e Banach algebras it is shown t h a t there is a one- to-one correspondence between the regular m a x i m a l ideals and the complex-valued homomorphisms in the sense t h a t every regular m a x i m a l ideal consists of the e l e m e n t s / , such t h a t

l*

0,

where /* is the functional, which gives the corresponding h o m o m o r p h i s m (L. 23 A). I f M denotes a variable regular m a x i m a l ideal and /* the corresponding functional in the above sense, then the function

/ (M) = 1" (1)

is called the Gelfand representation of the element / on the space of regular m a x i m a l ideals. As topology on this space we choose the weakest topology in which all the functions / ( M ) are continuous.

I n our case the regular m a x i m a l ideals are in one-to-one correspondence to the points xE G, since the functionals h a v e the f o r m /(x). I f we in this w a y identify the space of regular m a x i m a l ideals and G, L e m m a 1.51 implies t h a t the topology of the regular m a x i m a l ideal space is the original topology on G. The topological space G is therefore the topological space o] regular maximal ideals and every [unction is its own Gel[and representation.

F r o m the general theory of c o m m u t a t i v e B a n a c h algebras we get the following t h e o r e m (L. 2 4 A Cor., L. 25D). F o r the t r u t h of B it is essential t h a t the algebra is regular (L. 19F), and this is the f a c t in our ease because of L e m m a 1.24.

T t t E O R E M 1.51. A. I / /(x) EF, then

II/

nl

16 Y N G V E D O M A R

B. Let E be a subset o/ G and suppose that we have an ideal in F with the pro- perty that, /or any x o E E, it contains a /unction g (x) E F, such that g(Xo) * O. Then the ideal contains every / ( x ) E F, such that Af is compact and included in E.

Of f u n d a m e n t a l i m p o r t a n c e is the following theorem:

T H E O R E M 1.52. The elements /(x) with compact A / are dense in F.

Before we s t a r t the proof, we shall introduce a new concept, using a t e r m i n o l o g y from L. 31E.

D E F I N I T I O ~ 1.51. A /unction / ( x ) E F is said to have an approximate identity i/ /or every e > 0 there exists a compact neighborhood N o/ ~ with the property that /or every /o(X) E F o such that ]o(~) is non-negative, vanishes outside N and satis/ies

fo = 1,

N

we have

II/(x)

The proof of Theorem 1.52 will a p p e a r as a n e a s y consequence of the following l e m m a :

L E ~ M A 1.52. Every element /(x) E F with an approximate identity can be ap- proximated arbitrarily closely by elements o/ the /orm / ( x ) . g ( x ) , where g(x) 6 F and Ag is compact.

P R O O F O F L E M M A 1 . 5 2 . Choose an a r b i t r a r y s > 0 and a set N which gives

~ - a p p r o x i m a t i o n s of /(x) in t h e sense of Definition 1.51. Then we use the f u n c t i o n g(x), defined in L e m m a 1.23. We t h e n h a v e

II/- /.gll <-IIl- /.g ll + ll/ll, llg ll<- + .ll /ll,

which proves the lemma.

P R O O F OF T H E O R E M 1.52. The a b o v e lemm~ implies t h a t the closure of the elements with c o m p a c t A / c o n t a i n s all functions with an a p p r o x i m a t e i d e n t i t y . However, L e m m a 1.21 h a s the consequence t h a t all elements in F 0 h a v e an ap- p r o x i m a t e i d e n t i t y , and therefore t h e y are c o n t a i n e d in the closure. And because of A s s u m p t i o n I I e v e r y e l e m e n t in F is c o n t a i n e d in the closure.

Theorem 1.51 B with E = G and Theorem 1.52 h a v e the following i m p o r t a n t consequence (the Wiener T a u b e r i a n theorem, ef. L. 2 5 D , Cot.).

H A R M O N I C A N A L Y S I S B A S E D O N C E R T A I N C O M M U T A T I V E BA~qACH A L G E B R A S 17 T H E O R E M 1.53. Suppose that a closed ideal in F has t]~e property that it contains, /or every x o E G, a /unction / (x), such that / (%) ~- O. Then the ideal is the whole algebra F.

Or, u s i n g a l g e b r a i c t e r m i n o l o g y : Every closed proper ideal is contained in at least one regular maximal ideal.

C H A P T E R I I

Special Algebras and Special Elements

1. Various examples of Banach algebras F

A v e r y s i m p l e e x a m p l e of a n a l g e b r a F is t h e s p a c e of all c o n t i n u o u s f u n c t i o n s on G, v a n i s h i n g a t i n f i n i t y , if we a s n o r m c h o o s e t h e u n i f o r m n o r m . A s s u m p t i o n I is t r i v i a l t o verify, a n d A s s u m p t i o n i I is fulfilled a s a c o n s e q u e n c e of t h e w e l l - k n o w n f a c t t h a t f u n c t i o n s which are F o u r i e r t r a n s f o r m s of f u n c t i o n s in L I ( G ) are dense in t h e class. T h i s e x a m p l e s h o w s t h a t even if we c a n e x p r e s s t h e f u n c t i o n s in F 0 a n d F ' a s F o u r i e r t r a n s f o r m s of f u n c t i o n s on G, t h i s is in g e n e r a l n o t t r u e for all t h e e l e m e n t s in F .

H o w e v e r , t h e p a r t i c u l a r cases, w h e n t h i s is p o s s i b l e , are of g r e a t i n t e r e s t . T h e classical e x a m p l e is t h e space of f u n c t i o n s /(x), which are F o u r i e r t r a n s f o r m s of f u n c t i o n s f ( ~ ) E L t ( ~ ) a n d w i t h t h e n o r m

II!11= .r

G

B e u r l i n g [1] a n d W e r m e r [17] h a v e s t u d i e d on t h e r e a l line R m o r e g e n e r a l B a n a c h a l g e b r a s of F o u r i e r t r a n s f o r m s of f u n c t i o n s E L 1 (/~). (R d e n o t e s here as in t h e f o l l o w i n g t h e r e a l line u n d e r t h e u s u a l t o p o l o g y . ) T h e B e u r l i n g a l g e b r a s are s a i d t o be of n o n - q u a s i a n a l y t i e t y p e if for e v e r y n e i g h b o r h o o d N of t h e i d e n t i t y in R t h e y c o n t a i n a n o t i d e n t i c a l l y v a n i s h i n g f u n c t i o n which v a n i s h e s o u t s i d e N . T h e c o r r e s p o n d i n g s u b c l a s s e s of t h e W e r m e r a l g e b r a s are a l g e b r a s which s a t i s f y a c e r t a i n a s s u m p t i o n (A), [17] p. 538. The B e u r l i n g n o n - q u a s i a n a l y t i c a l g e b r a s are a p p a r e n t l y a l g e b r a s of t y p e F, a n d t h e s a m e is t r u e for t h o s e W e r m e r a l g e b r a s , which s a t i s f y (A), a p a r t f r o m a n u n e s s e n t i a l difference in t h e d e f i n i t i o n of t h e n o r m [17] p. 537 (6).

F o r a l g e b r a s of t h i s k i n d , i.e. a l g e b r a s which are d e f i n e d as c o n v o l u t i o n a l g e b r a s on t h e d u a l g r o u p , t h e v e r i f i c a t i o n of A s s u m p t i o n I A in 1.1 is o f t e n a v e r y d i f f i c u l t p r o b l e m . T h i s m a t t e r was d i s c u s s e d in t h e c i t e d p a p e r s a n d we shall i l l u s t r a t e i t

2 - - 5 6 3 8 0 2 . Acta mathematica. 96. I m p r i m 6 le 2 m a i 1956.

1 8 Y N G V E D O M A R

further b y discussing a n a t u r a l generalization of the Beurling algebras to an a r b i t r a r y locally c o m p a c t Abelian group G.

L e t :~(~) be a function on (~, measurable with respect to the H a a r measure, bounded on every c o m p a c t set and satisfying

(2) >_

for every ~ E G and

(kx + ~ ) -< P (~)" P (~) for every pair of points ~21 and ~2 in G.

Then consider the multiplicative B a n a c h algebra of functions

(2.11)

(2.12)

/(x)= ~ (x,~)/(~)d2,

G

where [(k) e L 1 (G), and where

G

D E F I ~ ~ T I 0 1~ 2.11. We denote this Banach algebra F {~}.

I t is e a s y to see t h a t the inequality (2.11) is necessary in order to h a v e (1.51) fulfilled, i.e. in order t h a t the algebra is of type F.

The algebra

F{~}

T H E O R E M 2.11. F ( p } is an algebra F i[ and only i[ [or every 20 C log [~(n~0) ] <

1 - - ~ . . . . ' ( 2 . 1 3 )

The proof will be given in 2.2.

I t m a y be pointed out t h a t the condition (2.13) is well known on the real line.

All questions of this kind are closely connected to notions of quasi-analyticity on the real line, a n d T h e o r e m X I I in P a l e y - W i e n e r [13] is a suitable tool in m a n y similar

c a s e s .

An especially interesting case of the space F{~} is the following. Suppose t h a t G = R , i.e. t h a t /~ m a y be represented as the real line - ~ < t < ~ . We choose

~(t)= ~ avlt[ ~,

0

H A R M O N I C A N A L Y S I S B A S E D ON C E R T A I N C O M M U T A T I V E B A N A C H A L G E B R A S 1 9

w h e r e {a,,}o is a sequence of n o n - n e g a t i v e n u m b e r s such t h a t t h e l e a s t n o n - i n c r e a s i n g m a j o r a n t of

r 1 6 2

O TM

1

is c o n v e r g e n t , a n d such t h a t for e v e r y m a n d n a,,+n(m+n)!<_arnm! .a,,n!.

The c o n d i t i o n (2.12) is e a s y t o v e r i f y , a n d ( 2 . 1 3 ) i s fulfilled a c c o r d i n g t o a t h e o r e m b y C a r l e m a n [6] p. 50. T h e r e f o r e t h e space is of t y p e F .

T h e i n t e r e s t of t h i s s p a c e lies in t h e f a c t t h a t we m a y c o n s t r u c t a v e r y closely r e l a t e d s p a c e in t h e following w a y . Consider on R t h e s p a c e of all f u n c t i o n s / ( x ) such t h a t

II111=

while

snp l l + < x / l + o ,

o Ixl~x0

w h e n x 0 - + ~ . This s p a c e is a m u l t i p l i e a t i v e B a n a c h a l g e b r a if II/d[ is c h o s e n as n o r m , a n d t h e f a c t t h a t t h e f u n c t i o n s in t h e a l g e b r a F{~b} are dense in t h i s new a l g e b r a c a n be u s e d t o show" t h a t i t is of t y p e F . W e will n o t go i n t o t h e d e t a i l s c o n c e r n i n g t h e p r o o f .

A p a r t i c u l a r case is w h e n a ~ = 0 if v is g r e a t e r t h a n a c e r t a i n i n d e x n. I n t h i s case we h a v e t o s u p p o s e t h a t /(~>(x) is c o n t i n u o u s a n d t h e a l g e b r a is t h e n s i m p l y t h e m u l t i p l i c a t i v e a l g e b r a of all f u n c t i o n s w i t h n c o n t i n u o u s d e r i v a t i v e s , v a n i s h i n g t o g e t h e r w i t h t h e d e r i v a t i v e s a t i n f i n i t y . I t is t h e n p o s s i b l e t o use as n o r m

II111 = sup I x l < ~ o

I1 +(x) I.

Since we h a v e i n t r o d u c e d in D e f i n i t i o n 1.51 t h e n o t i o n of e l e m e n t s in F w i t h a n a p p r o x i m a t e i d e n t i t y , i t m a y be s u i t a b l e t o c o n s t r u c t a s p a c e F w h e r e n o t all t h e e l e m e n t s are of t h a t k i n d .

W e f o r m t h e f u n c t i o n

1 P(t)-12~zlt]&(1 § It])

on /~. I t is e a s y t o s h o w t h a t if t 0 * 0

o~

P (t) p ( t o - t) dt <- :P (to).

c o

20 Y~GVE DOMAR L e t us t h e n c o n s i d e r t h e class of f u n c t i o n s

co

/ ( x ) = ] e - ~ t z [ ( t ) d t ,

- o o

w h e r e f(t) is c o n t i n u o u s e x c e p t p o s s i b l y a t t = 0 , a n d where f(t) = o@(t))

a t t = 0 a n d t = ~ . Choosing t h e n o r m

II111= sup f(t)

t~-O ~ - '

we g e t a n a l g e b r a F . H o w e v e r , a n e l e m e n t such t h a t [(t) is d i s c o n t i n u o u s a t t = 0 , can n o t h a v e a n a p p r o x i m a t e i d e n t i t y . F o r such a n e l e m e n t / ( x ) we h a v e f u r t h e r - m o r e t h a t , if a * 0, t h e f u n c t i o n

! ( x ) e ~ x

does n o t b e l o n g t o t h e class. Therefore, in g e n e r a l we d o n o t g e t new e l e m e n t s b y m u l t i p l y i n g a n e l e m e n t w i t h a c h a r a c t e r . T h i s f a c t a c c o u n t s for s o m e of t h e com- p l i c a t i o n s in t h e d i s c u s s i o n s in c h a p t e r s 3 a n d 4.

F i n a l l y we give t h e f o l l o w i n g space, which i l l u s t r a t e s t h e f a c t t h a t for a given v a l u e x 0~-0, t h e f u n c t i o n s [(x) a n d [ ( x + x0) n e e d n o t be e l e m e n t s of F a t t h e s a m e t i m e .

W e c o n s i d e r on R t h e class of f u n c t i o n s

co

/ (x) = /l (X) + / 2 ( x } = j" e ~tX fl (t) d t + .[ e-itz f2(t)dt,

. o o - o r

where / l ( t ) ( 1 + [ t l ) E L 1, f 2 ( t ) E L 1, a n d where /2(x) v a n i s h e s in t h e i n t e r v a l 0 _ < x _ < o o T h e n we p u t

Illll=inf { }r (1 +ltl)Ifl(t)ldt+

TIf~(t)ldt},

w h e r e we v a r y all t h e p o s s i b l e r e p r e s e n t a t i o n s of /(x). I t is v e r y s i m p l e t o s h o w t h a t we g e t a n a l g e b r a F .

T H E N E C E S S I T Y .

2. P r o o f o f T h e o r e m 2.11

L e t us a s s u m e t h e o p p o s i t e , n a m e l y tha~ for s o m e 2 0 E (]

log [ p ( n x o ) ] _ c~ , (2 21)

/ 7 2 1

H A R M O N I C A N A L Y S I S B A S E D O N C E R T A I N C O M M U T A T I V E B A N A C H A L G E B R A S 21 whereas for every n e i g h b o r h o o d N of the i d e n t i t y in G we h a v e a n o t identically vanishing f u n c t i o n

l ( x ) = ~ (x, 2)I(PC) d~, d

which vanishes outside N, a n d is such t h a t

J

5( )ll(Pc)l

G

We m a y then proceed in e x a c t l y t h e same w a y as in the proof of t h e o r e m 1.41 with the only difference t h a t ~ ( - 2 ) all t h e time is e x c h a n g e d to /)(2). I t is t r u e t h a t t h e inequality (1.46) uses the relation (1.44), which is n o t true for t h e f u n c t i o n /)(Pc), b u t (2.12) is a p p a r e n t l y sufficient to g u a r a n t e e t h a t ( 1 . 4 6 ) i s valid even if

~ ( - P c ) is e x c h a n g e d t o /)(Pc). W e therefore get for every c, such t h a t 0 < c < ~, t h a t there exists a f u n c t i o n

b(O)

satisfy a n d

cn= ~ e-~n~

e

Co~- 1 K

(2.22) (2.23)

for every integer n a n d for some finite c o n s t a n t K.

T h e reason w h y t h e p r o o f of t h e c o n t r a d i c t i o n in these relations causes us some trouble is t h a t there exists no exact correspondence in t h e t h e o r y of F o u r i e r seriea to t h e T h e o r e m X I I in P a l e y - W i e n e r [13] on F o u r i e r integrals, which v a n i s h on a hMf-line. The m e t h o d which we shall use is t o transfer t h e series into an integral with similar properties, a n d t h e n use the P a l e y - W i e n e r theorem.

S t a n d a r d a r g u m e n t s on t h e Fourier series in question show t h a t there is no real restriction to assume t h a t

I cn I < (2.24)

- o o

T h e n w e c a n p r o v e 0 < c < z ~ / 2 .

F o r every real n u m b e r y we define

an(y)

e - t y o

the contradiction, starting from a value of c, such t h a t

in - ~ _ < 0 < ~ , and p u t

dn (y) = an (y)" cn.

t h e Fourier coefficients of t h e

22 Y N G V E D O M A R

Apparently

[ dn (y) ] . i~ (n~o) <- d (2.25)

- o o

for some finite constant d, independent of y. In the interval - z / 2 _< 0 _< z / 2 we have dn(y) e~n~ f e ~v(o ~ ) . d b ( ~ p ) = 2 n e - ~ V O B ( _ y ) , (2.26)

c

where

B ( t ) = re-it~

- c

Using the Parseval relation, (2.26) and (2.23) we obtain for any integer n [ B ( n + y ) l = [ f e -'~~ ~VOdb(O) l

c

- - 2 2 " ~ m = - o r

cmdn-m(Y) B~-Y)

However, by (2.12) and (2.25)

(mxo) p ((n - m)Xo)ld.-m (U) l . . . . io(m~o) --m= ~ /~(n~o)'P(m~ o)

1 or d

<- ~ (n ~o) m =- ~ ~ b ((n - m)~o)ldn_ ~n (y)] <-- ~ (n ~o) and hence

K . d 1

[ B ( n +Y) l < - 2 ~ l B ( _ y ) l ' ~ ( n 2 o ) "

I t follows from (2.22) t h a t

for [Y] -< 5, if 5 is sufficiently small. Therefore K . d ] B ( n + Y) l <- fo(n~o )

if n is an integer and l yl_<(~. B y (2.24), b(O) is absolutely continuous and b' (0) E L 2 ( - c, c). Hence B (t) E L 2 ( - oo, oo), and the inequality above, together with the assumption (2.21), gives

; log ]B(t) l d t = - ~ .

3 1 + t 2

Using the cited theorem by Paley-Wiener we see t h a t this implies t h a t B(t)=-O, which is contradictory to (2.22).

HARMONIC ANALYSIS BASED ON CERTAIN COMMUTATIVE BAb!ACH ALGEBRAS 23 T ~ E S v F F I C I E N C Y. I n t h e proof of the sufficiency it seems difficult to avoid s t r u c t u r a i considerations. We shall start b y considering some cases when t h e g r o u p G has a v e r y simple s t r u c t u r e and t h e n step b y step e x t e n d t h e t h e o r e m to t h e general case.

The case when G is a discrete group D is trivial, since G is t h e n c o m p a c t (L. 38A), i.e. /5(5) is bounded. T h e cases w h e n G is t h e real line R or t h e u n i t circle S, b o t h under the usual topology, are obvious consequences of T h e o r e m X I 1 in P a l e y - W i e n e r [13].

N o w let G be t h e direct p r o d u c t G I • 2 of two g r o u p s G 1 and G3, for which the t h e o r e m is true. The points in G m a y be w r i t t e n in t h e form x = x l + x 3 where x l E G 1, x 3 E G 3. 0 is t h e n the direct p r o d u c t of 01 a n d 03, and we m a y therefore put 2 ~ 5 I+53, where 2 I E 0 1 and x~.E02 (L. 35A). This representation of G and 0 can he done such that always

(X, X) = (Xl, 51)" (X2, 52).

W e denote the identities in G, G1, G3, G1, a n d 0 3 b y o, ol, 03, 51, a n d 63, a n d assume t h a t the H a a r measure on G is n o r m e d in such a w a y t h a t it is the direct p r o d u c t of the H a a r measures on G~ and 0 3 .

Suppose t h a t

(5) = ~ (5~ + 53)

fulfills the requirements of t h e o r e m 2.11. L e t N be an a r b i t r a r y n e i g h b o r h o o d of o.

I t contains a n e i g h b o r h o o d of t h e f o r m N 1 X ~ u 2 where N 1 c G 1 a n d N 3 ~ G 3 are neigh.

borhoods of o~ a n d 03, because of t h e definition of t h e direct p r o d u c t topology. The functions

15 (5~) =/5 (51 + 63) a n d 15 (53) = 15 (5 s + 6~),

considered as functions on G1 a n d (~2, satisfy t h e conditions in T h e o r e m 2.11. A n d since t h e t h e o r e m was supposed to be true for these groul~s, we can for v = l , 2 find a function

1~ (x~) =

S

0 < I I ] . ( 5 , ) 1 ~ ( 5 , ) d ~ . < ~ . T h e n let us f o r m t h e function

/ (X) = [ (X 1 -~ T2) = fl (Zl)" f2 (X2)

= ~ (Zl, 51) (X2, 52) fl (51)* f3 (~t) d ~ i d52 = S (x, 5) fl (51) f2 (52) u s .

d, • o, 0

24 Y~GVE DOMAR I t vanishes outside N, b u t n o t identically, a n d satisfies

.f [ t 1 ( ~ 1 ) I , I t ~ ( & ) l ~ ( ~ ) d ~ ~

Ifl(~i)l If~(~)[ ~(&) b(~)d~

= .r I / ~ ( : ~ , ) l i , ( ~ ) < ~ l . . f i/~(&)l 7 , ( ~ ) d ~ < oo.

5, d.

Therefore, t h e t h e o r e m is true for t h e group G.

As a consequence of this the t h e o r e m is true for all groups of the form

R~•215

We are n o w in a position to prove the t h e o r e m for an a r b i t r a r y locally c o m p a c t Abelian g r o u p G.

L e t N be an a r b i t r a r y n e i g h b o r h o o d of the i d e n t i t y o in G. The multiplicative algebra of Fourier t r a n s f o r m s of functions E L I(G) u n d e r t h e usual n o r m is an algebra F. B y applying L e m m a 1.22

function

to this class of functions we see t h a t there exists a

g (x) = ~f (x, ~) 0 (Y) d~, G

where ~(~)E L 1 (G) vanishes outside a c o m p a c t s y m m e t r i c neighborhood 0 of 6, and such t h a t

g (o) = 1, (2.28)

while

[g (x) I -< 89 outside N.

We d e n o t e b y G1 the subgroup of G, which consists of all points ~1, included in some of t h e sets

n.C=C+C+...+C,

i.e. the group, generated b y C. This is a new locally c o m p a c t g r o u p u n d e r the induced topology, and we m a y as H a a r measure on G1 choose the restriction to G1 of the H a a r measure on (7. According to a t h e o r e m b y A. Well [16] p. 110, G1 is the dual g r o u p of a group G 1 of the t y p e (2.27), and hence the t h e o r e m is t r u e for the g r o u p Gx.

I f t h e function ib(~) satisfies the conditions in the t h e o r e m with respect to G, t h e n the same is true with respect to C:~ for the function P(Xl), defined as the re- striction of p(Y) to G 1. F o r t h a t reason we can find, for every n e i g h b o r h o o d N1 of the i d e n t i t y in t h e dual g r o u p G1, a function [(~1) on G1 such t h a t

HARMONIC A N A L Y S I S B A S E D ON C E R T A I N COMMUTATIVE BANACH ALGEBRAS 2 5

G~

while

5,

vanishes outside N 1. W e m a y as N 1 choose t h e subset of G 1 where I J" (xD ~1) g(s d ~ ] > 1 ,

d,

for this set is obviously open, and i~ contains the i d e n t i t y because of (2.28).

~ e e x t e n d t h e definition of [(~) to the whole of G b y defining [ ( ~ ) = 0 outside (~1, And since for e v e r y x E G t h e restriction to 571 of the functions (x, ~) are charac- ters on C~1, we see t h a t t h e a b o v e conditions i m p l y t h a t the function

G

vanishes w h e n e v e r I g (x) l ~ ~,I a n d hence it vanishes outside N. T h e condition (2.29) m a y be w r i t t e n

G

and since N was a r b i t r a r y , this p r o v e s t h e t h e o r e m in the general case.

3. The class q~

Basic functions in the h a r m o n i c analysis on R are the functions e i~x, where ), is a complex n u m b e r , a n d to some e x t e n t also the o r d i n a r y polynomials

P~ (x) = ~ a,~ x m.

rn--O

These two classes of functions h a v e correspondences on a n y locally c o m p a c t Abelian group, a n d we shall n o w s t u d y to w h a t e x t e n t these generalized exponentials and p o l y n o m i a l s on G belong to F. T o this end we shall m a k e the following definition.

D E F I N I T I O N 2.31. We denote by 09 the class o/ all /unctions on G, which coincide on any given compact set with .some /unction in F.

T h e following t h e o r e m shows t h a t r contains all generalized exponentials.

T ~ E O R E ~ 2.31. Suppose that ~(x) is a continuous /unction on G, satis/ying (X 1 -L X2 ) = g ( X l ) , 0r (;~2)

/or every x I and x 2 in G. T h e n ~(x) C (P.

26 Y N G V E D O M A R

P ~ o o F . I t is only necessary to consider t h e case when ~ ( x ) ~ 0 . L e m m a 1.24 t h e n shows t h a t it is possible to find a function / ( x ) E F', such t h a t

/ ( x ) ~ ( - x ) d x = 1.

AI

We denote b y g (x) the function which coincides with :r (x) on the c o m p a c t set, consisting of all points x = x 1 - x 2 , where x 1 E C a n d x 2 E AI, while it vanishes outside t h e set. Then if x E C

h ( x ) = f [ ( x o ) g ( x - x o ) d x o = S / ( x o ) c c ( x - x o ) d x o = : c ( x ) " ~ / ( x o ) : c ( - x o ) d x o = : c ( x ) ,

hr hf A r

a n d the Fourier t r a n s f o r m )~ (3) satisfies

(~.) = f(~.).

~ (~,),

where ~ (3) is continuous a n d bounded. Therefore, it follows from A s s u m p t i o n I C in 1.1 t h a t h E F .

R ~ M A R K. The question concerning the existence of other functions u (x) t h a n t h e b o u n d e d characters has been answered b y M a c k e y [12]. H e has f o u n d t h a t there exist unbounded [unctions o~ (x) it and only i/ there exist non-trivial continuous homo- morphisms o/ R into (~, i.e. one.parameter subgroups o/ G.

DEI~I~ITION 2.32. A continuous [unction P ( x ) on G is called a polynomial o/

degree n, i/ /or every x and x o in G

P (x + v xo)

is a polynomial o/ degree <_n, considered as a [unction of the variable non-negative integer v, while at least one o/ these polynomials is exactly o] degree n.

We m a y m e n t i o n as an example t h a t if :r is an u n b o u n d e d function of the kind described in Theorem 2.31, t h e n

{log (x)I}"

is a polynomial of degree n.

T H ~ O R E ~ 2.32. Every polynomial belongs to @.

We need t h e following l e m m a :

L ~ . ~ A 2.31. Let P~(x) be a polynomial o/ degree <n. Then [or any given "x o

V=O \ v / i8 independent o[ x.

9 I A R M O N I C A N A L Y S I S B A S E D ON C E R T A I N C O M M U T A T I V E BA[NACH A L G E B R A S 2 7

PlCOOF OF LEMMA 2.31. The above definition of a polynomial of finite degree can be given on a n y semi-group, and if we omit the continuity assumption, the semi-group need not even have a topology. I t will turn out from the proof t h a t the lemma is still true in t h a t general case if we make the extra assumption t h a t the semi-group is commutative.

Apparently it is enough to prove t h a t

Q (x, %) = Q (x + x 1, x0) (2.31)

for any given pair of points x and x 1 in G.

The expression

Pn(x 4- ktXl § VXo)=R(~, v)

has the property t h a t for every choice of non-negative integers /x, v, #0 and v o

R ( # + 2#o , v+ XVo) (2.32)

is a polynomial of degree <_ n in the non-negative integer variable 2. If we choose

# = V o = 0 , g o = l , and then v = / ~ o = 0 , v0= 1, it follows from the elementary theory of arithmetical series t h a t

R (#, v) = ~ ap, q/~" v ~.

P,q=0

B y choosing suitable values of /~0 and v 0 in (2.32) it is obvious t h a t the coefficients ap, q vanish whenever p + q > n. Then

~ ( - W R(~, v ) = ( - U~n! a0,n,

v=fl

which is independent of #. We obtain the two members of (2.31) by putting # = 0 and # = 1 in the above expression, and hence the relation (2.31)is true, which proves the lemma.

P R O O F OF T ~ E O R E ~ 2.32. We shall prove the theorem by induction.

Lemma 1.24 shows t h a t the theorem is true for polynomials of degree 0. Let us suppose t h a t it is true for polynomials of degree _ < n - l , and we have then only to prove t h a t the construction is possible for an arbitrary polynomial Pn (x) of de- gree n.

We choose a function [(x)E F' such t h a t

[. l ( x ) d x = l.

Af

2S Y~GVE DOMAR The function

P,~ i ( x ) = P n ( x ) - ~ / ( x l ) P n ( x - x l ) d x 1 Af

satisfies because of l e m m a 2.31

~ o ( - 1 Y

(n)

= Af

= Q (x, Xo) - ~[ / (xl) Q (x, xo) d xl = O Af

for every x a n d x0, a n d therefore it is a polynomial of degree _ ~ n - 1 .

N o w let C be the given c o m p a c t set, a n d let g(x) coincide with P~(x) on the c o m p a c t set, consisting of t h e points x = x 1 - x 2 , where x 1 C C a n d x 2E AI, while it vanishes outside this set. The function

h (x) = S / (x0) g (x - x0) d x 0 Af

belongs to F, as can be shown in t h e same w a y as t h e similar s t a t e m e n t in t h e proof of T h e o r e m 2.31. A n d we have for x E C

Pn 1 ( X ) = Pn (x) -- h (x).

B y a s s u m p t i o n we can find a function ]~(x)E F, coinciding with t h e polynomial P~_l(X) on C. Hence we get if x E C

P~ (x) = h (x) + k (x), a n d this proves t h e theorem.

Our definition of polynomials is quite different from t h e definitions of polynomials and generalized polynomials in t h e t h e o r y of distributions on locally c o m p a c t Abelian groups b y J. Riss [15]. The connection of our concept and his is n o t obvious, and a s t u d y of this problem seems to require extensive s t r u c t u r a l considerations. The a u t h o r hopes t h a t he will be able to r e t u r n to this subject.

CHAPTER I I I

T h e Spaces A a n d the S p e c t r u m 1. The spaces A

L e t F be a B a n a e h algebra of t h e kind described in c h a p t e r I a n d let A be a n o r m e d linear space. We assume t h a t to each / E F a n d each a E A there corresponds an element / o a E A a n d t h a t this correspondence has t h e following properties: