THE ALGEBRAS OF FUNCTIONS WITH FOURIER TRANSFORMS IN Lp: A SURVEY

A SURVEY

by

Ronald Larsen

Wesleyan University, Lliddletown, Connecticut

and

University of Oslo, Oslo

0. INTRODUCTION:Beginningin the early 1960's a considerable number of research papers have been devoted to the investigation of the algebras of functions with pth power integrable Fourier transforms. The intent of this article is to give an expository survey of this work together with some comments on generalizations and open problems. YTe shall attempt throughout to compare these algebras with the group algebras, indicating both the similarities and the differences. 0::.1ly a very fevv proofs will be given. Before we begin the exposition we shall set some notation and recall some well known results that will be needed in the survey.

If G is a locally compact Abelian topological group, then

"

G vvill denote the dual group, that is, the group of continuous characters on G _• The Greek letter ^),. and _1l will denote Haar measure on G and G ...

'

respectively. If G is compact, it will always be assumed that A. is normalized so that A.( G) = 1

'

^while

if G is discrete, then we shall assume the normalization sush that A([t}) = 1 , t E G. The topology on a topological group will always be assumed to be Hausdorff. Given 0 < p < co , the linear space of equivalence classes of complex-valued functions on G whose pth powers are integrable with respect to Haar mea- sure will be denoted by Lp(G) , and the linear space of equiva- lence classes of essentially bounded complex-valued Haar measur- able functions on G will be denoted by

Iz:o(

G) • If 1 ^~ p ^{~ CD} 9

then L (G) _p is a Banach space under the norm

11 .I £ II I p =

c J ^I

^{£ C t )}

^I

^P

d)~. ^c

^{t ) ) 1}

^/p

G

l!fliXl = ess sup! f( t)

I

t ~G

( 1 < p < oo) ,

(p

=

^{co ) •}

If (X~l,~) is a positive measure space, then Lp(~) 1_:::p~co,

will denote the analogous LP-spaces with respect to u •

The normed linear spaces of continuous complex-valued func- tions on a locally compact Abelian topological group G that are bounded, vanish at infinity, or have compact support will be de- noted by C ( G ) , C ( G ) , and

0 Cc(G) , respectively.

these spaces is, of course, the supremum norm llfll-v"\= supjf(t)! •

~~ t E G

The norm in

v7ith this norm the spaces O(G) and C₀ (G) are Banach spaces.

The convolution of two elements in L1 (G) , that is, f* g(t) = Jf(t-s)g(s)d),(s) ,

G

provides a multiplication operation in L1 (G) under which L1 (G)

is a commutative Banach algebra. The maximal ideal space of can be identified with G , and the Fourier (

"' =

^Gel'fand)

transform

f

of f in t 1^{(G) is given by} f(y)

=

Jc-t,y)f(t)dA(t)

G

(y E

G)

The Banach algebra L₁(G) is semisimple, regular, and Tauberian.

If G is a compact Abelian topological group, then Lp(G) , 1 < p

_:::::a ,

is also a commutative Banach algebra with convolution as multiplication. The maximal ideal space can once again be iden-

"

tified with G •

The Banach space of all bounded regular complex-valued Borel measures on G will be denoted by M( G) • The norm of u :in M( G) is the total variation of u , that is,

'll-lll

⁼ lui(G) • With con- volution of measures, that is,

u

*

v ( E ) = j r u ( E- s ) d v ( s ) , G

as multiplication, the space Ivi(G) becomes a semisimple commuta- tive Banach algebra with identity.

In general, if A is a commutative Banach algebra, then the maximal ideal space of A will be denoted by ~(A) , and the Gel'fand transform of X in A by X ^A If XcA then

A

X = [x

l

x E X} • The dual space of a Banach space B will be denoted by B* . All linear spaces are taken over the field of complex numbers.

The letters <V, JR., ~, and

r

will be used to denote the lo- cally compact Abelian topological groups of the complex plane, the real line, the integers, and the unit circle in the complex plane,

respectively. The symbol # will be used to indicate the comple- tio:n of a proof.

The basic notation~ terminology, and results of functional analysis and Banach algebra theory that will be used can all be found in Larsen '31,32], while the necessary results in harmonic analysis are available in Hewitt and Ross [18,19], Larsen [30], and Rudin [50]. The reader is referred to these references for further discussion of the preceding items and additional results that will be utilized in the sequel.

1 ~ B1'i.SIC PROPERTIES OF A (G). If G is a locally compact Abe- p

lian topological group and 0 < p ,::: oo , we define

If p = oo or G is discrete (equivalently, G is compact), then ...

it is apparent that Ap(G)

=

L1 (G) as linear spaces. These two situations will generally be ignored in the succeeding pages. In particular, we shall always assume that G is nandiscrete when discussing A (G)

p unless otherwise stated.

If 1 _::: p < :x::> , then on defining

!!f!!P _I = 'lf[l _{,, :1}

+lit!!

_{' 'p}

it is easily verified that A (G)

p is a commutative Banach algebra under the norm

ll·JIP

and with convolution as multiplication. Fur-

thermore, A (G)

p is a

n

·111-dense ideal in is evident on noting that the set

The denseness

is contained in A (G)

p and recalling that has an approxi- mate identity whose Fourier transforms have compact support [50~

p.51]. We remind the reader that a commutative Banach algebra A has an approximate identity if there exists a net [u }

1'(, contained

in A such that lima.!!xua.-xl!

=

0 for each x in A • The appro- ximate identity [ua} is bounded if there exists some M > 0 such

that supa.!lur:t,IJ ,::: M •

The maximal ideal space of

"

A (G)

p can be identified with Clearly every y in ^G defines a complex homomorphism F of A (G) by means of the formula

p

F(f) ⁼ f(y)

...

"

G •

Conversely, suppose F is a complex homomorphism of Ap(G) • We observe that if f is in Ap(G) , then for each positive integer n we have

< '~r-¹ n jlf'' +ll(fn-¹)"11 l!.fl!

'I 1 ^{I II} 1 ^{'I "CC 'I} p

< (jlfiJ )n-1\'f:IP

I 1 . I " •

Hence, for each n

vvhence, on letting n tend to infinity~ we conclude that

1 im c ^II

r

^{II P ) 1}

^{In .::: !}

^{l f}

ⁿ

1 • n

However~ a standard result from the theory of Banach algebras [32, p.81] entails that

!F(f)l < lim(l!fniiP) 1/n n

< !!f!l1

Thus each complex homomorphism of bounded complex homomorphism on the

A (G)

p is seen to be a

ll·!l

1-dense subalgebra

II ₁₁

.p

_{,J 1-}

A (G) p of L1 (G) • Consequently, F can be uniquely extended to a com- plex homomorphism of L1 (G) • Since ^{6 (} L _{1 (} G ) ) and G can be i-^,..

,..

dentified~ it follows at once that t:.(A (G))

p and G are identi- fiable as point sets. Some additional argument, together with the preceding development, establishes the following theorem. The de- tails are available in Larsen [30,pp.195-199] and Larseny Liu, and Wang [34].

THEOREM 1.1. Let G be a non discrete locally compact Abe- lian topological group. If ^{1 ,::} p < co , then Ap (G) is a Tauber- ian semisimple commutative Banach algebra under the norm

11·!1P

and with convolution as multiplication whose maximal ideal space

is

is homeomorphic with G • ,..

Obviously, if

II· •! ll I I q-dense in

1 < p < q < co , then A (G) c Aq (G) and A (G)

- p p

Aq(G) . It is almost as nobvious" that Ap(G)

J

Aq(G) • However, the proof of this latter fact is far from tri- vial for arbitrary nondiscrete locally compact Abelian topological groups G (contrary to the assertion in Larsen C30, p.197]), al- though fairly straightforward proofs exist for specific groups such as the unit circle

r

and the real line E . A complete proof of the distinctness of the algebras Ap(G) is given in Tewari and Gupta [55]. Prior to their work some partial results were proved in Larsen [30, p.208] and Martin and Yap [36]. The latter authors also showed that A1 (G) is a subset of A2 (G) of

category I.

An application of Plancherel'sTheorem [19, p.226] reveals that A2 (G)

=

L1 (G)" L2 (G) • This specific algebra has also been studied by Warner C57]. Larsen, Liu, and Wang [34] asserted with- oux proof that if 1 < p < 2 and 1/p+ 1/p'

=

1 , then L1 (G) n

L (G)

p is a proper subset of A ,(G) •

p The containment is estab- lished via the Hausdorff -Young Theorem [19, p.227], whereas the properness of the containment is not so evident. Lai [23] has shown that L1 (G) ⁿ Lp(G) is not only a proper subset of Ap1(G) , but it is also an !l·IJP '-dense subset of category I. His proof is a modification of Hewitt's proof [17] that Lp(G) is a proper

11·

!lp, -dense subset of Lp, (G) of category I whenever 1 < p < 2 , 1/p+ 1fp' = 1 , and G is infinite.

If 1 ~ p ~ 2 , then AP(G) is isometrically linearly iso- morphic to the dual space of a certain Banach space. This is a result of Liu and van Rooij [35]. 'Ne shall now describe the ap- propriate Banach spaces in more detail.

If X and Y are Banach spaces with norms

ll·llx

^and

ll·!Jv'

^..L.

respectively, then X x Y is a Banach space under the norm

II (x,y) II = llxllx

⁺

i!Ylly

Given a closed linear subspace

vr

of the Banach space X xY 9

we denote by X vW Y the quotient Banach space X ^X Y/W with the usual quotient norm. The theorem of Liu and van Rooij asserts that if

w

is the closure in C₀(G) x C₀

(G)

of the linear sub- space

then there is an isometric linear isomorphism of A1 (G) onto

... *

[ C _{0 (} G ) vW C _{0 (} G ) ] If 1 < p < 2 and

w

is taken to be the closure in d₀(G) x Lp, (G) of W , 1 /p + 1 /p' = 1 , then A (G)

0 p

is isometrically linearly isomorphic with [C (G) ₀

vw

^{L ,} _p ^{(G)]* •}

is, as usual, defined by f(t)

=

f(-t) , t E G • The element ^,..., f

We shall utilize this description of A (G) , 1 ~ p ~ 2 , when

p

we discuss the multiplies of A (G)

p in Section 5.

The dual space of A (G)

p has been described in Larsen, Liu, and Wang [ 34].

2. IDEAL THEORY. A fundamental result concerning the alge- bras is the relationship between closed ideals in

and in We have already observed that there exists a one- to-one correspondence between the maximal regular ideals in Ap(G) and in L1 (G) •

that 6 ( Ap ( G ) )

This is precisely the content of the observation

"'

can be identified with G • This correspondence between maximal regular ideals can be extended to all closed ideaw as seen from the next theorem.

We recall that a primary ideal in a commutative Banach alge- bra is a closed ideal that is contained in precisely one maximal regular ideal.

THEOREM 2.1. Let G be a nondiscrete locally compact Abe- lian topological group and let 1 ~ p < ~ •

(i) If

r

₁ is a closed (maximal regular2 primary) ideal in

L1 (G) , then I ⁼ I 1 n Ap(G) is a closed (maximal regular, primary) ideal in A (G)

p

(ii) If I is a closed (maximal regular, primary) ideal in Ap(G) and I 1 is the

!1·11

1-closure of I in L1 (G) , then I 1 is a closed (maximal regular, primary) ideal in L1 (G) and I=

I 1 n Ap (G) •

Proofs of this result in various degrees of generality can be found in Burnham [2], Lai [22], Larsen, Liu, and Wang [34], and Reiter [44, pp.129-130].

If one identifies 6(AP(G)) with

G

= 6(L 1 (G)) , then given closed ideals I in Ap(G) and I 1 in L1 (G) such that I=

I 1

n

AP(G) it is evident that h(I) = h(I 1 ) and k[h(I)] =

k[h(I 1 )] nAp ^{(G) •} Here, of course, h( I) denotes the hull of the ideal I and k(E) denotes the kernal of the set E of maximal regular ideals ~ 32, p. 160]. This remark, combined with Theorem 2.1 and well known results for L₁(G) [32, pp.173,184,197,199, and 217], yields at once the following corollary:

COROLLARY 2.1. Let G be a nondiscrete locally compact Abelian topological group and let 1 ~ p < co •

(i) The algebra Ap(G) is_§ re&ular cQ_mmutative Banach algebra.

(ii) A closed ideal I in A (G) is m~~imal regular if and p

only if I is primary.

(iii) If G is compact, then every closed ideal I ~ Ap(G) is of the form

...

I= k[h(I)] = [f!f EAP(G), f(y) = 0, y Eh(I)} •

(iv) If G is noncompac~ then there exists some closed ideal I in A (G)

p such that I f k[h(I)] •

(v) If I is a proper closed ideal in A (G) , then there

p

exists a maximal regular ideal J in AP(G) such that J ~ I • It is also easily seen that I is a closed ideal in Ap(G) if and only if I is a closed translation invariant linear sub- space of A (G)

p I is a closed linear subspace of Ap(G) such that ,-sf E I whenever f E I and s E G • The translation operator ,.s is defined by the formula ,-sf(t) = f(t-s) 9 t E G .

If A is a commutative Banach algebra we remind the reader that the Qroblem of spectral synthesis is tbe question of deter- mining when a closed ideal I in A is the kernel of its hull, that is9 when is I = k~(I)] . Equivalently, i t is the question of determining which closed sets E in ~(A) are the hulls of precisely one closed ideal I in A In other words, given a closed set E in 6(A) , when is i t the case that h(I) = E im- plies I = k(E) ? A closed set E in 6(A) with this latter property is called a set of spectral synthesis. The reader is re- ferred to [32, pp.192-194] for further details. Theorem 2.1 and Corollary 2.1 show that the problem of spectral synthesis is es- sentially the same in both A (G)

p and

One of the standard abstract results dealing with the problem of spectral synthesis in a semisimple regular commutative Banach algebra A is Silov's Theorem [32'Y 9 p.206]. The theorem provides a sufficient condition for a closed set in 6(A) to be a set of spectral synthesis whenever the algebra A satisfies some ad- ditional restrictions commonly lonovvn as Ditkin's conditions [32, p.204]. A number of different proofs have been given showing that

'Y

A (G) satisfies Ditkin's condition, and hence that Silov's Theorem

p

is valid for Ap(G) . See, for example, Burnham r3], Lai (24], Larsen [29], and Yap [63].

Some more specialized results concerning spectral synthesis in A (G)

p have been studied by Larsen [28] and Warner [57]. The latter author discusses only the algebra A2 (G) . We say that a

A

linear subspace X of Lp(G) is invariant under characters if (s,•)g EX whenever g EX and s E G • The symbol (s,•) na-

,..

turally denotes the continuous character on G determined by s

"

in G • If X is a closed linear subspace of Lp(G) that is in- variant under characters, then

is evidently a

J(X) = [f!fEA (G), fEX}

p

closed ideal in A (G) p

.

On the other hand, if I is a closed ideal in A (G) p and H (I) p denotes the closed linear

A ,.. ,..

subspace of Lp(G) spanned by I

=

^{[f!f E I} 3} then Hp(I) is in- variant under characters. The next theorem is proved in Larsen

[28] and Warner r57] for p

=

2 . We denote the closure and interior of a set E by cl(E) and int(E) , respectively.

THEOREM 2.2. Let G be a nondiscrete locally compact Abel- ian topological group and let 1 _::: p < ,-:::o •

(i) If X is a closed linear subspace of Lp(G) that is in- variant under characters₂ then J(X) is a closed ideal in Ap(G)

such that J(X) = k(h[J(X)]) •

(ii) If I is a closed ideal in A (G) such that I = k[h(I)]

p

and h(I)

=

cl(int[h(I)]) , then I

=

J[HP(I)] .

Theorem 2.2 is actually valid in a more general context.

Indeed⁹ let A be a semisimple regular comrnutative Banach algebra and suppose ~ is a positive regular Borel measure on 6(A) such that }..l(E) > 0 for each open set E contained in 6(A) _• Let A (\.l)

'

^{1 < p < ·XJ '} denote the

p set of ^X in A such that x ^A is

in Lp(!.l) _• Then Ap(\.l) is a Banach algebra under the norm

Theorem 2.2 has a valid analog for A _p· (u) provided A contains an approximate identity [un} contained in A₁(\.l) for which

\!ua.ll = 1 for each a. . This and other results conJerning Ap ( 11) are discussed in Larsen [28].

Dietrich [7] has established some interesting results cancer- ning the closed ideals I in A (G)

p that are countably generated.

His main result for arbitrary nondiscrete groups is the next theo- rem.

THEOP..Em 2.3. Let G be a nondiscrete locally compact Abel- ian topological group and let 1 _:: p < ^{ry:] •} If I is a closed countably generated ideal in A

(&) ,

then h(I)

p - is a clopen set

"

in G .

The proof of this theorem is rather long and the reader is referred to r7] for the details • Some consequences of the theorem are more easily established.

COROLLARY 2.2. Let G be a nondiscrete locally comEact Abelian topological group and let 1 _:: p < ^{CD •}

(i)

If G is connected, then no nonzero proper closed ideal ^"' in A (G) is countably generated.

p

(ii) ted.

(iii)

If G is connected~ then A (G)

p is not countably genera- ....

If G is noncompact and E is a closed subset of G that is not a set of spectral synthesis for AP(G) , then any closed ideal I in Ap(G) such that h(I) ⁼ E is not countably genera- ted. In particular, k(E) is not countably generated.

(iv) If I is a closed ideal in Ap(G) that is countably gen- erated, then I = kCh(I)] •

(v) If G is noncompact and E is a closed subset of G that "' is not a set of spectral synthesis for A (G) , then there exists

p

an uncountable family [Ia} of distinct closed ideals in Ap(G) such that h(Ia) = E for each ~ and no Ia is countably gener- ated.

PROOF. We shall sketch the proofs of parts (i) and (v). In the first case, suppose I is a closed countably generated ideal

A (G) h(I) ^"

in _p Then, since is clop en and G is connected:~ we h(I)

= ¢

^{h( I)}

^{= G}

^{h(I) "}

see that either or _• If

=

^{G 9 then ob-}

viously I

=

^{[0] as A} _p ^(G) is semisimple, whereas ^{i f} h( I)

= ¢ ,

then by Corollary 2. 1 (v) we concl11de that I

=

A p (G)

.

Thus there exists no nonzero proper closed countably generated ideals in AJG) •

To prove part (v) we let E c G ^.... be a closed set that is not a set of spectral synthesis for L 1 (G) • Such a set always exists as G is noncompact [32, p.199]. From a result of Helson [19, p.

601] we know there exist uncountably many distinct closed ideals such that h(J ) = E .

a Let I a = J a n A p (G) • Then is an uncountable family of distinct closed ideals in A (G) _p for which h(I )

=

E •

f'J. Moreover, none of the I can be countab1y

a

generated since E is not a set of spectral synthesis for L1 (G) ⁹ and so 9 by Theorem 2. 1 9 not a set of spectral synthesis for A _p (G)._,.

·;r

Dietrich [7] also contains a precise description of the closed countably generated ideals in Ap(G) when G is compactly gener- ated [18, pp.35 and 90].

THEOREM 2.4. Let G be a compactly generated nondiscrete locally compact Abelian topological group 9 let 1 < p < ro , and let I be a nonzero closed ideal in Ap(G) • Then the following are eq_uivalent:

(i) The ideal I is coQntably generated.

(ii) There exists a nonnegative integer n , a compact Abelian

"'

topological group K 9 and a cofinite subset E of K suuh that G

=

;;zn x K and

I = [f! f E A (G),

f (

y) = 0, y E r><E } •

p

All of these results concerning countably generated ideals in Ap(G) are also valid for L1 (G) • This situation was investiga- ted in Dietrich r6].

Our discussion of the ideal structure of Ap(G) has so far emphasized the similarities with the ideal structure of L1 (G) • However, there are some marked differences to which we now wish

to turn our attention. First we shall recall some general defi- nitions and some results concerning L1 (G)

.

Let A be a commutative Banach algebra. An ideal I in A is prime if xy E I implies either X E I or y E _{I •} A mapping

-X-

.

o A ^_, A is an involution if for any x9y in A and any a. ⁱⁿ

cv we have

(a) (x+y) * = x* + y

*

(b) (xy)* = x*y*

.

(c) (ax)* = ^{(J,X -)!-}

(d) (x-l")*

=

^X

The algebra A is a *-algebra if A has an involution such that

!lx* l1 =

!lxll, x E A • A positive linear f1mctiona1 w on a Banach

algebra A with involution is a linear functional on A such that w(xx*) ~ 0 , x E A .

The algebras L₁(G) and AP (G) , 1 ,:: p < oo , are Banach *- algebras. Indeed, it is easily verified that the formula f*(t)

=

f(-t) , t E G , defines an involution on L₁(G) and AP(G) such that !!f*l!₁

=

l!fl!₁ and llf-:<!lp

=

f!fl!P

Porcelli and Collins [41,42] proved that if I is a maximal ideal in L1^(G)

'

^{then I} is regular, prime, and closed, and if I is a prime ideal in L1^(G) ' ^{then I} is maximal if and only if I is closed. Furthermore, a theorem of Varopoulos [19, p.270]

concerning Banach *-algebras with bounded approximate identities entails that every positive linear functional on L₁(G) is con- tinuous. £11 of these results fail to be valid for Ap(G) as shovm by Martin and Yap [36].

THEOREM 2.5. Let G be a nondiscrete locally compact Abel- ian topological group and let 1 :::, p < C8 •

(i) The algebra A (G)

p contains a maximal ideal that is neither regular, prime, nor closed.

(ii) There exists a positive linear functional on Ap(G) that is discontinuous.

These results for the algebras L1 (G) and Ap(G) concerning ideals and positive linear functionals are, with the exception of the use of the theorem of Varopoulos, all consequences of a lemma due to Porcelli ~40, p.88]. It is worth while stating this lemma explicitly as i t will enable us to easily see why the preceding results are valid.

If A is a commutative Banach algebra, then we denote by [A2 ] the ideal in A generated by A2 = {x;)T! x,y E A} •

PORCELLI'S LElY.lMA. (a) Let A be a commutative Banach alge- bra such that CA2 ]

f

^{[O} •} Then the following are equivalent:

(i)

(ii) The algebra A contains a maximal ideal that is not prime.

Moreover, if [A2 ]

f

A , then each nonprime maximal ideal in A is a maximal linear subspace of A that contains [A2 ].

(b) Let A be a commutative Banach algebra without identity and let I be a maximal ideal in A . Then the following are equivalent:

(iii) The ideal I is regular.

(iv) The ideal I is prime.

The results about maximal ideals in L1 (G) that were men- tioned before Theorem 2.5 follow at once from Porcelli's Lemma on recalling that L1 (G)-* L1 (G) ⁼ [L 1 (G) ] ^{2 =} L1 (G) . The crucial

step needed in applying Porcelli's Lemma to Ap(G) is the fact that CAP(G) 2 ] is a 1\·]JP-dense proper ideal in Ap(G) whenever G is nondiscrete. Once this result is kno~n, then part (i) of Theorem 2.5 is an immediate consequence of Porcelli's Lemma.

Theorem 2.5 (ii) is established via a straightforward construction.

Namely9 if I is a non prime maximal ideal in A (G) p

'

^{then I is}

a ll·lJP-dense maximal linear subspace of A (G) p that contains [Ap(G) 2 ]

.

^{If f} ₀ ^{E Ap(G) ,..., I}

_'

then every f E A (G) p can be -vvri tten uniquely in the form f

=

g + a.fo 9 for some g E I and

a. E ^<D _• Define

td

f)

=

^a. _' ^{f E A (G)} _p

.

^{Clearly w} is a linear

functional on Ap(G) 9 and ^{;) is positive because f * ^f~(- _E Ap(G) 2 c I whence w(f*f*)

=

⁰

.

The functional t:.J is discontinuous since it is not identically zero, but does vanish identically on

the

II· !i

P -dense linear subspace I •

The fact that L1 (G) * L1 (G)

=

L1 (G) can be proved by an ap- peal to the well known factorization theorem of P.J. Cohen [19, p.270]. This theorem implies that if A is a commutative Banach algebra with a bounded approximate identity, then A 2

=

^A

.

^The

question of the existence of approximate identities and factori- zation for the algebras Ap(G) will be taken up in the next sec- tion.

3. APPROXIMATE IDENTITIES AND NONFACTORIZATION. As we hinted at the close of the preceding section, it is well known that L1 (G) has bounded approximate identities [32, pp.109 and 110]. On the other hand, since Ap(G) *Ap(G) c [Ap(G) 2 ]

f

^AP(G) it is apparent from Cohen's factorization theorem that AP(G) does not have a bounded approximate identity. The main results concerning factor-

ization and approximate identities for A (G)

p are collected in the next theorem.

THEOREM 3.1. Let G be a nondiscrete locally compact Abel- ian topological group and let 1 < p < co .

(i)

Ap (G) •

The ideal

( i i ) Ap ( G )

*

^{Ap (} G )

f

^{Ap ( G ) •}

(iii) (Ap(G) 2 ] c Amax( 1^sP-1 )(G) • (iv)

~A 1 ^(G)

²

^J

^{c At(G) •}

11 _II .I!P _IJ -dense proper ideal in

(v) The algebra A (G)

p does not have a bounded approximate identity.

(vi) The algebra Ap(G) has an approximate identity [u } ,.. Cl

such that {u Cl} c

c

^{c (G) •}

The proofs of parts (i)-(v) can be found in Martin and Yap [36], while explicit proofs of part (vi) are available in Lai [22] and Larsen [30, p.199]. Other proofs of the nonfactorization of Ap(G) have been given by Burnham [4] and Wang [56]. Burnham's argument is particularly elegant and depends on the following sim- ple lemma whose proof is an application of the generalized H5lder inequality 118, p.138].

LEMJ'.lA 3. 1 • Let (X ,-.3, ^1...1.) be a positive measure space with

!J. (X) > 0 , and suppose for some p , 1 < p < co.

If there exists a positive integer n ~ p and f 1 ,f2 , ••• ,fn in

Lp(~) such that f = f 1f 2 ••• fn, then f ^{E L}1 ^(~) •

Burnham actually proves the lemma under the assumption that f has the indicated decomposition for every n such that n ^~ p • However, this stronger assumption seems to be unnecessary.

Burru1am's nonfactorization results in [4] can be summarized as follows:

THEOREM 3.2. Let G be a non~iscrete locally compact Abel- ian topologic~~ group~~ let X be a subset of M(G) such that

,.. ^"

X -~X c X and X

¢.

_{L1 (G) •}

(i) If for some _{p ' 1 <} p <::a, then ^{v .L}.. ·•}

(ii) If X c L (G) for some p

'

¹ < p < ²

'

^{then X *X}

I

X

.

p

-

(iii) If X c L₁ (G) :, Lp (G) for some p

'

¹ <p<CD, then X-><-X

I

^X _•

PROOF. We shall prove only part ( i). If X* X

=

X , then given a positive integer n ^~ p there evidently exist ^~₁ ^,~₂^, ••

in X such that Consequently, ^"' ~

=

and vk E ^"' Lp (G) , k ^"'

=

1 , 2, ••• , n • Hence, by Lemma 3.1, we conclude that ~ ^E L₁(G) , whence

X

^{c L}1

(G)

contrary to assump-

tion. Therefore • 'fr, ^I'

Obviously Theorem 3.2 (i) implies at once that A (G).x-A (G)I p p A (G)

'

¹

p < p <::o. If 1 < p ^<~ and G is any nondiscrete group, then part (iii) of the theorem shows that ( L ₁ (G) '1 Lp (G) )

*

(L₁(G) !!Lp(G))

I

L₁(G) '1Lp(G), and if G is an infinite compact group, then LP(G)

*

Lp(G)

I

Lp(G) • The latter results have been established previously by Yap [61] and Edwards [8], respectively.

" ,..

The hypothesis that X¢ L1 (G) is necessary for the validity of Theorem 3.2 as can be seen by letting X be the set of trigo-

A A

metric polynomials on a compact Abelian group. Then X c L1 (G) and X* X = X •

We might remark in closing that i t is apparently not known for commutative Banach algebras A if A2 = A implies A has a bounded approximate identity. Empirical support for this conjec- ture is provided by the foregoing development and the work of Wang [56].

4. FUNCTIONS THAT OPERATE IN Ap(G). If A is a commutative Banach algebra, then a complex-valued function F defined on an open subset E of ill is said to operate in A if F(x) E A A

whenever x E A and the range of ~ is contained in E • It is a well known result of the theory of Banach algebras that essen- tially every analytic function operates in a commutative Bauach algebra [32, p.152]. (In the case that A is without identity one needs to assume that F is analytic on some open set contain- ing zero and that F(O)

=

⁰ It is easily seen that this restric- tion can also be imposed, without loss of generality, even when A

....

has an identity, since in this case A contains the constant func- tions.) In particular, analytic functions operate in L1 (G) and AP(G) . For L1 (G) a good deal more is known. Before we sum- marize these results for L1 (G) we need a few additional notions.

A complex-valued function F defined on an open subset E of the Euclidean plane

m

² is said to be real-analytic on E if to every point (s₀,t_{0 )} in E there exists a power series expansion

of F with complex coefficients of the form

x;

F( _s, t) ₌ r' _- a _mn (s-s _o )m¹_' t-t _o )n m,n=o

that converges absolutely for all points (s,t) in some neighbor- hood of (s₀,t_{0 )} • A complex-valued function F defined on a

closed subinterval E of lli is real-analytic on E if F is real-analytic on some open planar set containing E •

Modulo some technical reductions, the next theorem gives a complete description of the functions that operate in L1 (G) [50, pp.132-147].

THEOREM 4.1. Let G be a locally compact Abelian topological group and suppose F is a complex-valued function defined on the closed interval i-1,1] such that F(O) = 0 .

(i) If G is an infinite compact group, ~ F operates in L1 (G) if and only if F is real-analytic on some neighborhood of zero.

(ii) If G is noncompact, then F operates in L1 (G) if and only if F is real-analytic on [-1,1] •

Portions of this result have been extended to the algebras Ap(G) by Lai [27]. The proof is patterned after the one for L1 (G) •

THEOREM 4.2. Let G be a noncompact nondiscrete locally compact Abelian topological group, let 1 < p < "XJ , and suppose F is a complex-valued function defined on the closed interval [-1,1]

such that F(O) = 0 • If F operates in AP(G) 9 then F is real-analytic on C-1,1].

in

Little seems to be known about the functions that operate A (G)

p when G is a compact group. However, it is not dif- ficult to see that these f~~ctions need not be real-analytic. For example, we have the following result~

THEOREM 4.3. Let G be an infinite compact Abelian topolo- gical group, let 1 ~ p < 2 , and suppose F is a complex-valued function defined on ^(jj

.

If there exists a constant M > 0 '~

r ' ^{1 < r}

-

^{< p}

'

^{and some 8 > 0} such that [ F( z)

I ~

M!z!p/r when- ever

I

z

I

^{< 8} then F operates in A p (G)

PROOF. Let f E AP(G) and let K be a compact (= finite) subset of

G

such that

!f(y)!

^{< e ,}

y f

K. Then

r,..!F(t)(y)lr =

yEG

<

r !F(f) (y)

!r

⁺ r !F(f) (y)

lr

yEK yfK

I. "' lr

!"

^!P

r ₁ F ( f ) ( y ) ₁ + Mr f ( y ) • ,

yEK YiK

A A A A

from which we conclude at once that F(f) E Lr(G) as f E Lp(G) • However, since 1 < r < 2 , we deduce via the Rausforff-Young Theorem [15, p.227] that there exists some g E Lr,(G) _{c L 1}(G) , 1/r+1/r' ⁼ 1 , such that

follows immediately that

"' "

g

=

F(f) • Since A (G) c A (G)

r p

F operates in Ap(G) • #

it

Thus, for example, if h is a complex-valued function de- fined on <C that is bounded in some open neighborhood of zero, and ¹ < r < p ~ 2

'

then F(z) = !zlp/rh(z) operates in Ap(G) whenever G is compact. In particular, it is apparent that such functions need not be real-analytic.

If p

=

2 and G is compact~ then we can appeal to a result

of Rider [46] and the fact that A2^{(G) = L}2^(G) to obtain a com- plete description of a certain class of functions that operate in

THEOREM 4.4. Let G be an infinite compact Abelian group and let F be a complex-valued function defined on ~ • Then the following are equivalent:

(i) The function F operates in A2 (G)

(ii) There exist complex numbers ~ and ~ and a complex- valued function h defined on (f) that is bounded in some O:Qen neighborhood of zero such that F(z) = a.Z+f3Z+Iz!h(z), z E <C.

We close this section by describing a result on a problem analogous to that of characterizing the functions that ope1ate in A (G) . If G is an infinite compact group, then i t is obvious

p

that A1^(G)

=

L1(G)A . Consequently, by Theorem 4.1 (ii), if F is a complex-valued function defined on [-1,1] such that F(O)

=

0 , then F(f) E A₁(G) for every f E A₁(G) for which -1 ~ f(t)

< 1 , t E G, if and only if F is real-analytic on [-1,1] •

This observation has been extended somewhat by Rudin [49]. To be precise, we have the following theorem~

THEOREM 4.5. Let G be an infinite compact Abelian topolo- gical group and let F be a complex-valued function defined on the closed interval [-1,1] such that F(O) = 0. If F(f) EAP(G) for some p , 1 ~ p < 2 , whenever f E A₁(G) and -1 ~ f(t) ~ 1 , t E G , then F is real-analytic on [-1,1] •

in

Note that the function F in Theorem 4.5 does not operate A (G)

p

5. THE 1lliLTIPLIERS OF Ap(G) • Let A be a semisimple commuta- tive Banach algebra. A mapping T: A ^-+ A is a multiplier of A if T(xy)

=

(Tx)y

=

x(Ty) for all x9y in A It is known that every such mapping is actually a continuous linear transformation on A and that the space of all the multipliers of A , denoted by M(A) , is a commutative Banach algebra with identity with ope- rator composition as multiplication and the operator norm

\1 Til

=

sup ^1J Tx11

x EA

l!xl!=1

(T E M(A))

Furthermore, for each T in M(A) there exists a unique bounded continuous function ~~:, defined on 6 (A) such that (Tx)

=

^{cpx ' "}

for each x in A Moreover,

\lcp!!

00.:S !!Til Conversely, it is evident that if c~ is a bounded continuous function on 6(A) such that ... "

cpx E A whenever x E A , then the equation (Tx) ₌

cpx '

^"

x E A , determines a multiplier T of A • We shall denote by 'Yrl_(A) the algebra under pointwise operations of all the bounded continuous functions ^cp on 6(A) that correspond to multipliers of A • Proofs of these and other results concerning M(A) as well as a discussion of multipliers for topological linear spaces are available in Larsen [30].

A considerable amount is known about the multipliers of cer- tain corrrmutative Banach algebras. We shall describe some of these results for when G is arbitrary, and for L (G) , 1 < p

p

<CD, when G is a compact group, in order to give some context for our discussion of the multipliers of Ap(G) • We begin with

the most familiar result, namely the characterization of the multi- pliers of L1 (G) ^~30, pp.2 and 3].

THEOREM 5.1. Let G be a locally compact Abelian topologi- cal group and let T: L₁(G) ~ L₁(G) • Then the following are equivalent:

(i)) The mapping T is a multiE}ier of L₁(G) •

(ii) The mappinei T is a continuous linear transformation such that TTs

=

TsT , s E G .

(iii) There exists a unique 1-1 in M(G) such that Tf

=

^1-L*f

f E L1 (G) •

,.. ...

(iv) There exists a unique \.l in M(G) such that (Tf)

=

1-Lf,

Moreover, the correspondence between T and 1-1 defines an iso- metric algebra isomorphism of M(L_{1 (G))} onto M(G) .

Thus when A = L_{1 (G)} we see that :M(L₁ (G)) can be identi-

M(G)

*

'hl(L_{1 (G))}

fied with = C₀ (G) , whereas can be identified with M(G)

...

_•

If G is a compact Abelian topological group, then Lp(G),

1 < p < co , is a commutative Banach algebra under convolution and

M(Lp(G)) can also be identified with the dual space of a Banach space of continuous functions as we just noticed was the case for M(L1 (G)) The description of the appropriate Banach space is, however, not as simple as for L1 (G) •

Let G be compact and let 1 < p < 8.-) • Then we denote by A~(G) the family of functions h in C(G) of the form

1/p + 1/p ¹ = 1 , and such that

On setting

I^{I 11} it can be shown that ^{.t •} :iA::p AP(G) is a Banach space. p

p

is a norm on under which

The multipliers of Lp(G) can be identified with the dual space of A~(G) • More precisely9 we have the following theorem:

THEOREM 5. 2. Let G be a compact Abelian group. If 1 < p <x ~

then there exists an isometric linear i~rphism of M(Lp(G)) onto AP(G)* .

p

The isomorphism is the mapping cr: M(L (G)) ~ AP(G)* deter-

p p

mined by

a(T) (h) = _k=1 2:: Tf_{1 {} -l~ g1 ( 0) _{

- s

(Th)~(y)d~(y)

"

G

The theorem is due to Fig~-Talamanca [10]. The idea of the construction can be carried further to describe more general mul- tiplier algebras for Lp-spaces as done in Figa-Talamanca [10]

and Fig~-Talamance and Gaudry [11]. A discussion of these results is also available in Larsen C30, pp.129,180-190].

It is also possible to express the multipliers of Lp(G) in terms of convolution9 but this time convolution with objects more general than measures -namely pseudomeasures. However9 it is not generally the case that a pseudomeasure defines a multiplier. It

is, of' coUJ::"se, obvious that every measure !J. in M(G) determines a multiplier of Lp(G) by means of convolution. In order to make more precise sense of the preceding sentences we need to discuss pseudomeasures briefly.

It is well known, for any locally compact Abelian topological group G that L1(G) .... is a Banach space under the norm

n£n =

!lf:l 1 ,

f

_{E L1}(G) .... [32, p,134]. The dual of this Banach space, de- noted by P(G) , is called the space of pseudomeasures on G • If G is infinite, then M(G) ~ P(G) • Given a in P(G) and de- fining F

a by

it is not difficult to verify that Fa is a continuous linear functional on L1 ^{(G) •} Since L1 (G)""- can be identified with L~(G) we deduce the existence of a unique ~ in L:o(G) such that

a(f) =

J f(y)3(-y)dn(y)

....

G

....

This mapping from P(G) to L:o(G) is a surjective isometric linear isomorphism. The function ^{A a} is usually referred to as the Fourier transform of a • Once this Fourier transform is in hand it is easy to introduce a multiplication in P(G) so that it becomes a commutative Banach algebra. Namely, if a and v are in P (G) , then a

*

v is defined to be the unique pseudo- measure such that

=

^... O"V • This convolution product in P(G) reduces to the usual convolution for elements of M(G) • Further discussion of pseudomeasures is available in Larsen [30, pp.97-101].

We can now give a partial description of M(LP(G)) in terms

of pseudomeasures.

THEOREM 5.3. Let G be a compact Abelian topological group and let 1 < p < ^,XJ. If T E M(Lp(G)) ~ then there exists a

unig_ue pseudomeasure cr E P (G) such that Tf ⁼ a* f , f ^E Lp (G) • Ivloreove:;-, the correspondence between T and a defines a con tin- uous algeb~ isomorphism of ^l\~(L (G))

p into P(G) • If p

=

2 ⁹ then the correspondence is a surjective isometric algebraisomor- phi sm.

It is immediately apparent, either from Theorem 5.3 or

Plancherel's Theorem, that tvuL0 (G))

=

^C(G) when G is compact.

c...

Furthermore, we remark that if G is compact, then T is a multiplier of Lp(G) if and only if T is a continuous linear transformation that commutes with translation.

With these examples of multiplier theorems in mind, we now wish to see what is known about the multipliers of Ap(G) • For arbitrary nondiscrete groups not much information is available, although we do have the following result:

THEOREM 5.4. Let G be a nondiscrete locally compact Abel- ian topological group, let 1 < p < ^{·X' ,} and 1 e t T : A (G) ... A (G)

p p

(i) If T E M(Ap(G)) , then there exists a unig_ue pseudo- measure a E P(G) such that Tf =a* f , f E A1 (G) • Moreover, the correspondence between T and a defines a continuous alge- bra isomorphism of M(Ap(G)) into P(G) •

T

(ii) The mapping T is a multiplier of is a continuous linear transformation on

Ap(G) A (G)

p

if and only if such that

INe note that~ in general~ not every pseudomeasure determines a multiplier of Ap(G) • This will be evident from the next theo- rem which gives a complete description of M(Ap(G)) when G is noncompact.

THEOREM 5.5. Let G be a noncomp~~nondiscrete locally compact Abelian topological group, let 1 ~ p < ~ , and let T: A (G) .... A (G) • Then the following are equivalent:

p p

(i) The mapping T is a multiplier of AP(G) .

(ii) There exists a unique u in M(G) such that Tf = u* f ,

(iii) There exists a unique in M(G) such that " ,...,.

(Tf) = 1-1f,

Moreover, the correspondence between T and u defines an iso- metric algebra isomorphism of M(Ap(G)) onto M(G) .

PROOF. The proof in its full generality is due to Figa-

Talamanca and Gaudry r12]. It will not be given here. The reader is referred to either the original paper or to Larsen r30, pp.204- 206] for a complete discussion. However, there exists a simple and elegant proof that part (i) implies part (ii) in the case p ~ 2 , and we shall sketch the argument. The idea is due to Forelli.

If

p;::,

2 ·' then it is readily seen that M(Ap(G)) cM(A2 (G)), and so we may restrict our attention, without loss of generality, to _{A2 (G)}

=

_L1 ^{(G) I!} L2 (G) • If ^{T E} M(A2^{(G)) ,} then by Plancherel's Theorem r19, p.226] we have

I!Tf!l1 < 11Tf!l 2

<

IITII 'lfll

²

"'

= !IT!I<!!f!! 1 ⁺

!1£!1

_{2 )}

=

i!T!!('!f!i 1

⁺

1!flJ 2)

Now a well known result for noncompact groups G [30, p.78]

asserts that if f E LP(G) , 1 ,::: p < ^'XJ, then

1 im! If + ,. f 11 = 2 1

I

P !\ f 11

s

_,dJ

s p , ··p •

Hence, for each f E A2 (G) we have, on the one hand, limi'T(f+ Tsf) ¹¹1

=

lim1\Tf+ TsTf!! 1

=

^2i!Tf11₁

S ->X S .... ::::0

and on the other hand,

Combining these observations with the preceding inequality we deduce that

Iterating this process n times reveals that

and letting n tend to infinity vve conclude that

Thus T defines a

!! • !!

1- bounded linear transformation on the

11·'1

₁-dense linear subspace A2 (G) of L1(G) that commutes with translation, that is, ~ determines a unique element of

M(L 1 (G)) • The existence of the desired ^~ in M(G) now follows at once from Theorem 5.1.

The situation when G is compact is considerably more frag- mentary and less satifactory. If ¹ < p ~ 2 , then an easy appli-

cation of Plancherel's Theorem reveals that 11l(A (G)) _p = C(G) • In terms of pseudomeasures the result is described in the next theorem [25,30, p.207].

THEOREM 5.6. Let G be an infinite compact Abelian topolo- gical group. If ^{1 ~} p < 2 , then there exists an isometric alge- bra isomorphism of M(A (G))

p onto P(G) •

The isomorphism is naturally just the mapping indicated in Theorem 5.4 (i), namely, the correspondence determined by the for- mula Tf ^{= 0}

*

f • Incidentally, this formula now holds for all f

If p > 2 , then things become considerably more complicated.

~ A

In particular, we note that M(G)

t

$h((Ap(G))

f

^{C(G) • The con-}

tainments are obvious, but the fact that they are proper requires some argument. Indeed, we claim first that there exists some

for each ~ E C (G) , then the set

0

(Ap(G))₀

=

[flf E Ap(G),~f EAP(G) .... for all cpE C₀(G)}

is clearly equal to Ap(G) • However, it is known that (AP(G))₀

p > 2 , which contradicts the fact that the spaces A (G) _p are all distinct as pointed out in Section 1. Thus •r!((AP (G))

f ^c

^{(G.) •}

To show that M(G)Af ¹01(AP(G)) , p > 2 , we need to recall the notion of a Sidon set and some facts about such sets. Given

"' a compact Abelian topological group G a subset E of G is called a Sidon set if to every bounded continuous function

w

de- fined on E there corresponds a ~ in M(G) such that ~(y)

= w(y)

⁹

y

E E • For example, if G

= r

and

G =

~

'

then E =

[2n In= 1,2,3, ••• } is a Sidon set. In general, the dual of any infinite compact Abelian group always contains an infinite Sidon set. A general exposition of Sidon sets is available in [19, pp.

415-449;50, pp.120-130].

One additional fact about Sidon sets will be needed. If E c

G

is an infinite Sidon set and ~ E

C(G)

is any function

...

that vanishes off of E , then ~ E M(G) if and only if ~ E

...

L2 (G) . A proof can be found in Larsen ^~30, p.85].

Now suppose p > 2 is given, let m = p/2 , n = m/(m-1) , and choose r such that 0 < r < 2 and rn > 2 • Let E c G ^...

be an infinite Sidon set and let ~ E

C(G)

be any function such that

(a) ~(y) = 0

' ^y

f.

E

.

(b) ^I: _I·:P(Y)! 2 = ^{-v-. ~.}

yEG

(c) ^I:

lep(y)

lrn <-:a •

yEG

If f E A (G) , then by Holder's inequality we have

p

Thus ~f ^E _L2

(G)

c Lp(G) , and so ~ ^E M(G)

...

because E is a Sidon set and

f

^{'}1(.( Ap (G) ) •}

'»j_(Ap (G)) • cp

t

L2 (G) •

However, ~

t

Hence M(G)

It is possible, however, to give a description of M(A (G)),

p

p > 2 , as the dual of a certain Banach space of continuous func- tions. To be precise, if G is an infL~ite compact Abelian topo- logical group and p > 2

'

then for each T E M(A (G)) p and f E A₁(G) we set

r·

...

8(T)(f) = ^I (Tf) (·y)dn(y) J "

G and we define

These definitions make sense as M(Ap(G)) c M(A₁(G)) . Then

11 • _.I

f!

_{i B} is a norm on and we denote the normed linear space

p

so obtained by Bp(G) . It is evident that B(T) E Bp(G)* for each T E M(Ap(G)) , and we have the following theorem [30, p.212]~

THEOREM 5.7. Let G be an infinite compact Abelian topolo- gical group. If 2 < p < oo , then there exists a continuous line- ar isomorphism of M(A (G))

p onto B (G)* •

- p