FINITE DIMENSIONAL CONVOLUTION ALGEBRAS

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FINITE DIMENSIONAL CONVOLUTION ALGEBRAS

BY

E D W I N H E W I T T and H E R B E R T S. ZUCKERMAN 1

Introduction

Thc notion of convolution (Ger.

Faltung,

Fr.

produit de composition)

is a venerable one in m a t h e m a t i c a l analysis. The convolution

1 t"

~ . ~(~)

sin ( n + 8 9 1 6 2 1 6 2 1 6 2

2sin 89

is found in Dirichlet's original m e m o i r [12] on Fourier series, and similar convolutions are extensively used in the classical literature on Fourier series and integrals (see for example R i e m a n n [34]. Weierstrass's original proof [47] of the celebrated approx- imation theorem bearing his n a m e utilizes a certain convolution. Fractional integration and differentiation are defined b y means of convolutions (see for example Z y g m u n d [51], pp. 222-225). A perusal of a n y adequate t e x t b o o k on Fourier series or integrals will show the i m p o r t a n t pIace occupied b y the notion of convolution in the field of harmonic analysis. The Hilbert transform is of course a convolution. The classical theory of this transform has recently been extended by Z y g m u n d and Calderon [52].

More recently, it has been recognized t h a t measures and certain classes of a b s t r a c t linear functionals can be convolved. F o r functions of finite variation on ( - ~ , + ~ ), for example, see Bochner [5], pp. 64-74, and, from another point of view, Beurling [4]

and Gel'fand [15]. The notion is discussed a n d utilized

in extensv

b y Jessen a n d Wintner [22]. L. Schwartz has studied convolutions for the class of linear functionals 1 The authors wish to acknowledge financial assistance by the National Science Foundation U.S.A. during the preparation of this paper.

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6 8 E D W I N H E W I T T A N D H E R B E R T S. ZUCKERMAI~

called distributions ([3~], Ch. VI). Finally, we shah see t h a t the algebras discussed in Bourbaki [6], pp. 110-115, call be considered to be convolution algebras.

The importance of convolutions in the theory of Lie groups was recognized by H. Weyl (Peter and Weyl [50] and Weyl [49], Ch. I I I , w167 12-15). A very general description of convolution of linear

operators

is found in A. Weil's treatise on topological groups ([48], pp. 46-48), and it is to this source t h a t the present paper owes its original inspiration. The notions sketched b y Well have been utihzed and extended by m a n y writers (see e.g. Segal [39], Godement [16], and Buck [7]).

I n studying numbers of examples of convolutions and in particular analyzing the ideal structure of certain algebras in which multiplication is defined by a convolution, the authors have been led to formulate a very general, purely algebraic, definition of convolution algebra, This definition includes all of the examples of convolutions which we have found in the literature. 2 A preliminary announcement of a special case is found in Hewitt and Zuckerman [19]. Our definition includes as non- trivial cases both infinite and finite dimensional algebras, and in studying these two classes of convolution algebras, entirely different techniques are called for. In- finite dimensional convolution algebras are best treated by analytic and topological methods, while the tools of classical algebra are required in the finite dimensional case. The present paper is devoted primarily to a discussion of finite or at least finite dimensional objects, although when a theorem about an infinite situation can be obtained at no extra effort, we do not hesitate to state it. Some infinite dimensional examples are also included, in 1.4. A second communication will be devoted to infinite dimensional convolution algebras.

We use the following notation:

K denotes the complex number field;

~ n denotes the algebra of all n>< n complex matrices;

Zp denotes the zero algebra over K of dimension p ;

A Q B

denotes the direct sum of algebras A and B;

Kn denotes the direct sum K O " " O K with n summands.

Throughout this paper, functions and linear functionals are always complex- valued. Linear spaces and algebras are always over the field K ; a n d homomorphisms and ideals of algebras are taken in the algebra sense.

It does not include the composition of functions of 2 variables discussed in VOLTERRA a n d

P~.R~S [45], but this composition in its general form can hardly be called a convolution.

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FINITE DIMENSIONAL CONVOLUTION ALGEBRAS 69 w 1. Basic definitions and theorems

We begin with a number of definitions.

1.1. Definition. A non-void set G is said to be a semigroup if there exists a binary operation defined for all x, y e G (usually written as xy) such that x ( y z ) = (xy)z for all x, y, z e G. 3 The cardinal number of a finite semigroup is called its order, and is often written as o(G).

1.2. We list a few examples of semigroups.

1.2.1. A n y group.

1.2.2. A n y non-void set G, with x y = y for all x, y e G .

1.2.3. A n y non-void set G; a a fixed element of G, and x y = a for all x, y e G . 1.2.4. A n y non-void set G completely ordered under a relation _<, with x y =

= m a x (x, y) for all x, y~ G.

1.2.5. The set {a, a-t- l, a + 2, ...}, where a is a non-negative integer, and the semigroup operation is ordinary addition.

1.2.6. I n Appendices 1 and 2, tables of all semigroups of orders 2 and 3 will be found. Such tables have been computed independently by Carman, Harden and Posey [8] and b y Tamura [44]. Their results for orders 2 and 3 agree with ours.

Carman, Harden, and Posey have an incomplete table of semigroups of order 4. G . E . Forsythe [14] has computed the semigroups of order 4 b y mechanical means.

1.3. Definition. Let G be a semigroup, and let 3 be a linear space of functions, with the usual definitions of sum and scalar multiplication, defined on G. We suppose that

1.3.1. for all x e G and /~ 3, the function ~/, defined b y the relation z/(Y)= /(xY), 4 is an element of 3.

Now let I: be a linear space of linear functionals defined on 3. For L ~ 1 : , / e 3 , and x e G , let L y ( / ( x y ) ) denote L ( J ) . We suppose further t h a t

1.3.2. for a l l L e l : and ] e 3 ~ the function on G whose value at x is L ~ ( / ( x y ) ) , is an clement of 3 ;

1.3.3. for all L, M e l:, the linear functional N on 3 defined by the relation N ( / ) = M x ( L y ( / ( x y ) ) ) is an element of s Under these conditions, we write N as a An extensive discussion of abstract semigroups and related systems may be found in DUBREIL [13], Chapter II.

The function ]x is defined analogously: it has the Value ] (yx) at the point y.

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70 E D W I N H E W l T T A N D H E R B E R T S. Z U C K E R M A N

M , L ,

and call

M L*

the

convolution

of the linear functionals M and

L.

The linear space 1: is said to be a

convolution algebra.

Definition 1.3 includes all of the notions of convolution which the authors have found in the literature, whether of functions, measures, or distributions. 5

1.4. Examples of convolution algebras. We now give a few examples of both finite and infinite dimensional convolution algebras, showing the diversity of structures included under our definition of convolution. A close study of these examples, however, is not essential for understanding the remainder of the present paper.

1.4.1. Let G be the semigroup defined in 1.2.3. Let ~ be a n y linear space of functions on G containing the function identically equal to unity (written as 1). Let l: be a n y linear space of linear functionals on ~ containing the functional ~a such that

~ta(/)=/(a) for all )r I t is easy to see t h a t conditions 1.3.1-1.3.3 are satisfied and t h a t

L M = L ( 1 ) M ( 1 ) ~ a*

for all

L, MeF~.

1.4.2. Let G be a finite group. The group algebra of G is often described as the set of all formal complex linear combinations of elements of G, ~ ~ x, with term-

x e G

wise addition and scalar multiplication and with product defined by ( ~ :r ~ flu Y)=

x s G y s G

= ~ ~ ~xflyxy.

This algebra is isomorphic to the convolution algebra consisting of

x e G y s G

all linear functionals on the space of all functions on G. To see this, let 2, be the functional such t h a t

2, (f) = / (a),

for all

aeG.

Then

2~2~(/)=/(ab),*

as we shall show in the proof of Theorem 1.7. Hence 2a*~b=)~ab. Since the linear functionals

~a (a ~ G) form a basis for all linear functionals in question, the asserted isomorphism is established. I t m a y also be noted t h a t ~

~ a x f l u x y =

~ ( ~ z ~ , f l v ) z; this

z ~ G y e G z e G y e G

identity shows the isomorphism of the algebra defined here with the algebra ~:x (G) described in 1.4.6

in]ra.

1.4.3. A quite different example is provided by the algebra of all sequences of

a ~ ~ b

complex numbers { ,~}n=o. We write a = {a~},=0, b = { ~},=o, and so on. The element

a+b

is

{a~+b~}~o, ta={ta~}~_o,

for all

t e K ,

and the p r o d u c t a * b of a and b is defined by the relation a * b b~ k i l o This algebra is a convolution algebra

: oak - j = .

in the sense of 1.3. Let ~ be the space of all functions on N o = (0, 1, 2, 3 . . . . ) which vanish except on finite subsets of N o . As noted in 1.2.5, N O is a semigroup under 5 Group algebras of finite groups have of course been studied for coefficients lying in arbitrary fields (see for example v. D. WAERDEN [46], Ch. XVII). Other generalizations are mentioned in BER- MAI'r [2] and PAIGE [28].

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F I N I T E D I M E N S I O N A L C O N V O L U T I O N A L G E B R A S 71 addition. Let s be the space of all linear functionals on ~. I t is clear t h a t for all

a r162 =

A s IZ, there is a unique sequence { .}n=0 as above such t h a t

A (]) ~ an/(n)

for all

n = 0

a ^{o o}

] s ~ .

Conversely, every sequence { ~}.=0 defines a linear functional on ~. I t is easy to verify 1.3.1 for the present function space ~. To verify 1.3.2, let / be an arbi- t r a r y element ~ 0 in ~, and let p - 1 be the greatest integer such t h a t / ( p - I ) = 4 = 0 . Then ] ( m + n ) = 0 for all n e N o and m>~p. T h e r e f o r e A , / ( m + n ) = ~ a ~ / ( m + n ) = 0

r~=0

for all m_> p, and hence 1.3.2 holds. Condition ] .3.3 is automatically satisfied, since s here consists of all linear functionals on ~. L e t e, be the function such t h a t

e~ (m) = 5 ~ (n, m s No).

Then clearly A (e~) = an. F o r elements A and B of s we therefore obtain A * B b y computing

A B* (en)

for all n e N 0. We have

**A*B(e~)=Ak(Bz(e~(k+l)))= ~ ~akbz~,k~,= ~ akb~= ~akb~_k.**

k = 0 l = O k + l = n k,=0

a c~

Therefore the multiplication defined above for sequences ( ~}~:0 is actually convolution in the sense of 1.3.

1.4.4.'Consider the space ~x(T), consisting of all Lebesgue integrable functions on the circle group T. F o r /, g s ~ I ( T ) , the integral

2 ~

/ g (x) = f / ( e ~(~-y)) g(e iy) dy,*

0

the convolution of / and g, defines a function which is again in s

1.4.5. F o r

[, ge ~I(R),

where R denotes the additive group of real numbers, we have

/ g (x) = / [ (x - y) g (y) dy,*

- o r

and this function / * g is again in ~I(R).

1.4.6. To show t h a t the operations described in 1.4.4 and 1.4.5 are convolutions in the sense of 1.3, consider an a r b i t r a r y locally compact group G. As ~, we t a k e

~ r (G), the space of all continuous functions on G which are arbitrarily small outside of compact sets, and normed b y I I / l l = m a x

I](x)I.

As s we take the space ~o~(G), con-

x e G

sisting of all bounded linear functionals on ~ ( G ) . I t is obvious t h a t 1.3.1 holds for

~ o ( G ) : if I / ( y ) l < s for y n o n e A , t h e n I ~ / ( y ) l < e for y n o n s x - l A , and

x - l A

is compact if A is compact. To verify 1.3.2 for the present case, we m u s t use

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72 E D W I N H E W I T T A N D H E R B E R T S. Z U C K E R M A N

F. Riesz's representation theorem for elements L of ~:r (G):

L(/) = f / ( x ) d2

(x), where

G

2 is a complex-valued, countably additive, bounded, regular Borel measure on G. 6 Consider the total variation 121 of ~t (see H e w i t t [18] for a definition). Since I~tl is regular, for every ~ > 0 there exists a compact subset B of G such t h a t I ~ I ( B ' ) < ~ . F o r an arbitrary

[e~r

and ~ > 0 , let A be a compact subset of G such t h a t I/(Y)[ <~] for all y e A ' . Then we have

I t is plain t h a t

f / (x y) d ]t (y) = f / (x y) d ]t (y) + f / (x y) d,~ (y).

G B B"

If/(xy)

d ~

<y) l ~ I1111. I ~ I(B')< ~ II/II.

B'

If x is not in the compact set

A B -1,

then we have

f /(xy)d2(y) ~_fl/(xy)ldlAl(y)~_~121(G),

B B

since

x y s A

and

y s B

imply t h a t

x e A B -1.

I t follows t h a t

f/(xy)d,~(y)

becomes

G

arbitrarily small outside of properly chosen compact sets. To show t h a t

f/(xy)d~.(y)

a

is continuous as a function of

x,

it is necessary only to note t h a t / is uniformly continuous. To verify 1.3.3 for the present ease, we observe t h a t

**IM*L(I)I=]f fl(xy)d~(y)~,(x) -< / fll(xy)l~l~l(y)dl,i(x)**

G G G G

<-II/ll I]tl ( G ) I # I (G)=II/II'IIL]IIIMll 9

Hence

M L*

is a bounded linear functional and

IIM~LII<_IIMll 9 IILll.

The convolution algebra ~ r (G) will be studied in detail in a second communication. This algebra has a subalgebra (actually a 2-sided ideal) consisting of all functionals for which the corresponding measures are absolutely continuous with respect to right H a a r measure on G. As is well known, for such measures /~, we have

d~u(x)=m(x)dx,

where m ~ C I ( G ) and

dx

is the differential of right H a a r measure. L e t /~ and v be two such measures, with

d r ( x ) = n ( x ) d x .

Then, writing the functionals in question as M and N, respectively, we have

F o r a l l s p e c i a l t e r m s u s e d i n t h i s p a r a g r a p h , s e e L o o M I s [ 2 4 ] .

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F I N I T E D I M E N S I O N A L C O N V O L U T I O N A L G E B R A S 73 M * N (/) = f f / (x y) n (y) d y m (x) d x = f f / (x y) m (x) d x n (y) d y

G G G G

= f f / ( x ) m ( x y -1) d x n ( y ) d y : f / ( x ) f m (~y-1) ~ ( y ) d y d x

G G G G

T h a t is, the measure corresponding to the functional

M N*

is absolutely continuous, and the differential of this measure is

f m ( x y - 1 ) n ( y ) d y d x .

This shows t h a t the

a

classical convolutions mentioned in 1.4.4 and 1.4.5 are actually convolutions in our present sense.

1.4.7. L e t the semigroup G be the additive group of real numbers, R. As the function space ~, we take the space of all continuous functions / on R such t h a t l i m / ( t ) and lim

/(t)

exist and are finite. Denote these limits b y E+ (/) and E_ (/),

t--~ e~ t - - ~ - ~

respectively. The space ~ is a Banach space under the usual addition and scalar multiplication and with

II/ll=sup I/(t) I.

I t is clear t h a t E+ and E_ are bounded

t e B

linear functionals on ~. As the space /:, we t a k e all bounded linear functionals on ~.

I t is not difficult to show t h a t every L e 1: has a unique representation of the form

oo

L (/) = f /(x) d~ (x) + ~ E_ (1) + fiE+ (/),

- o o

where 2 is a countably additive, complex-valued, bounded Borel measure on R and g, fle K. Conditions 1.3.1-1.3.3 are established b y a routine calculation, which we omit.

Hence we are in possession of another convolution algebra. Interesting features of this algebra are t h a t is has non-zero Jacobson radical (Jacobson [21]) and is non- commutative, in spite of the fact t h a t the basic group R is commutative. I n fact,

E+E_ (/)=*

lira ( lim

(/(x+y)))=

lim (E_ ( / ) ) = E _

(/),

X-.),O0 y = - - 0 0 ~--~ 0o

while

E_E+*

( / ) = E + (!).

We now give a few simple but basic results.

1.5. Theorem. E v e r y convolution algebra is associative.

Proof. L e t L, M, N be elements of the convolution algebra C, and let / e ~. Then

L ( M , N ) (/) = Lx (M * N~ (/(xy))) = L~ (M~ (N, (/(x*

(uv)))))

= L~ (M~ (N~ (/((xu)

v)))).

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74 E D W I N H E W I T T A N D H E R B E R T S. ZUCKERMA-N

On the other hand,

( L * M) * N (/) = L * M w (-Vv

([ (w r))) = L~ (Mu

(N~, (! ((x u) v)))).

This proves the present theorem.

1.6. Deiinition. Let G be an arbitrary finite semigroup, with elements Xl, x 2 . . . X n.

L e t

51 (G)

denote the linear space of all functions defined on

G,

and let I: 1 (G)denote the space of all linear functionals defined

on 51 (G).

Let ~ be the function in 51(G) such t h a t ~ ( x j ) = 5 ~ j ( i , j = l , 2 . . . . ,n) and let

;ti be the element of I:I(G ) such t h a t 4 i ( ~ j ) = J i j (i, j = 1, 2 . . . n). (Note the slight change in notation from 1.4.2.) Let !i, j! be the integer such t h a t

xixj=xc~,j 1

(i, j = ], 2 . . . n).

1.7. Theorem. ~ 1 ( G ) is a convolution algebra. I t is isomorphic to the algebra of all formal complex linear combinations ~ 0r x, where

.~eG

( .~ ~ x ) + ( ~ fl~x)= ~ (~+fl~)~, r( Y ~ x ) = =~ (,/~)x.9

x e G x e G x e G x e G x e G

and

( ~ ) ( ~ f l ~ y ) =

_{x e G} _{y e G} x e G y e G

~ ~ f l ~ x y .

ProoL The first statement of the present theorem is obvious. To prove the second, we observe first t h a t the functions ~i form a basis for ~I(G): if ! e 5 1 ( G ) , then ! = ~ ! ( x t ) q i . Next, the functionals 4i form a basis for I:t{G): if L ~ s ),

i=1

' t h e n L = ~ L ( ~ ) 4 ~ . I t is clear that 4i([)=[(x~) for all [ e ~ l ( G ) and i = 1 , 2 . . . n.

iffil

We shall now show t h a t 4~*4j=4t~,j] ( i , j = l , 2 . . . n). We use the identity

/ (xy) = ~ / (x~ x~) ~0k (x) 901 (y),

k , l = l

which is valid for all / e 51 (G) and x, y e G. We then see that.

4i.~: (4j.~ (/(xy))) = 4t.z ( ~ / (xk xz) q)k (x) 4s (~l))

l . k = l

l , k = l l , k = l

I t follows at once t h a t 4~-4,=4t~,, r Therefore, under the mapping ~ ~ q 4 ~ ~ ~x~,

t=1 t=1

we have an isomorphism between the algebra s (G) and the algebra described in the present theorem.

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I~INITE DIMENSIONAL CONVOLUTION ALGEBRAS 75 1.8. Remark. I t is obvious from t h e preceding theorem t h a t ~] (G) for a finite semigroup G is commutative if and only if G is commutative. This property no longer holds in the infinite dimensional case, as example 1.4.7 shows.

1.9. Theorem. Let G be a finite semigroup and let / e ~1 (G), and M, L ~ ~ 1 (G). Then

M~ (Ly (/(yx))) = Lx (My (/(x

y))). ~ Proof. Since

/ (xy) = ~ ~ / (x, x s) cfi (x) qJs (Y),

i j

we have

Lx (My (/(xy))) = ~ ~ / (x~ xs) L (cf~) M (Cps).

On the other hand, we also have

/(yx)= ~ V~ /(xuxv)~u(y)q~v(x),

u v

and hence it follows that

i x ( L y ( / ( y x ) ) ) = Z Z/(xuxv)L(cf~)i(q~v)= ~ ~ /(Xsxs)L(cf~)Mfffs)=Lx(My (/(xy))).

u v i J

1.10. Theorem. A finite dimensional algebra A is isomorphic to an algebra s (G) for some finite semigroup G if and only if A has a basis which is closed under multiplication in A.

Proof. If A is isomorphic to l: 1 (G) for some finite semigroup G under an isomorphism /~, then the elements /~-1(),1) . . . /~ 1 (),n) are a basis for A of the kind required. Conversely, if A possesses a basis which is closed under multiplication, say

al, a 2 ... a,,

then the elements al, a 2 .. . . an, form a semigroup under the multiplication operation in A. I t is clear t h a t A is the set of all sums ~_ cqa~, and t h a t

i = l

( ~ ~ a , ) ( ~ f l s a s ) : ~ ~fljaiaj.

Hence A is an algebra of the sort described in

i=1 j i , j = l

Theorem 1.7 and is accordingly isomorphic to an l:l-algebra.

1.10.1. The preceding theorem shows that the algebras described by Bourbaki [6], pp. 110-115 are convolution algebras, for finite semigroups. An extension to the infinite case offers no difficulties.

1.11. Theorem. E v e r y finite dimensional algebra A is a convolution algebra.

Proof. Adjoining a unit to A if necessary and using the regular representation, we obtain a faithful representation of A by complex p X p matrices, where

This theorem is due to Dr. THELMA CHA.NEY.

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.76 E D W I N H E W I T T A N D H E R B E R T S. Z U C K E R M A N

p___ dim A + 1. Thus we m a y regard ,4 as a subalgebra of ~l~p. Now, ~ p is not an l:l-algebra, as we shall show in 4.3, b u t ~ v O K is an l:l-algebra (this will be shown in Theorem 4.2). Therefore A is isomorphic to a subalgebra of an l:1-algebra. De- finition 1.3 makes it clear t h a t every subalgebra of a convolution algebra is a convolution algebra, and this completes the present proof.

1.12. Remark. Theorem 1.11 shows t h a t finite dimensional convolution algebras are too general to be of a n y interest from our present point of view. Therefore we shall limit ourselves here to a s t u d y of algebras of the form 1:1 (G) for finite semigroups G. These algebras are not nearly so general a class as one might at first suppose, and a certain a m o u n t of success has been obtained in determining their possible structures. F r o m time to time, we shall permit ourselves to m a k e statements regarding infinite semigroups, b u t we shall restrict ourselves to facts which can be obtained b y essentially finite arguments.

w 2. Finite sem|mrou~

2.1. I n spite of the large literature devoted to the algebraic theory of semigroups, we have found a n u m b e r of a p p a r e n t l y new theorems (in particular structure theorems), which are useful in classifying finite dimensional l:l-algebras. I n addition, we find it convenient to reformulate a few well-known ideas. The present section is devoted to this program.

We first give 5 simple theorems showing various methods of adjoining new elements to an a r b i t r a r y semigroup.

2.2. Theorem. L e t G be a semigroup. L e t z be an object not in G. Then G U {z}

(also written as Gz), with the multiplication rules

x y = x y as in G for all x, y e G , x z = z x = z for all xeG~,

is a semigroup. (G~ is said to be obtained from G b y adjoining a zero.)

Proot. I f u, v, w e G , then u ( v w ) = ( u v ) w b y hypothesis. If u, v, weG~ and a t least one of u, v, and w is z, then u ( v w ) = ( u v ) w = z .

A slight v a r i a n t on 2.2 is the following.

2.2.1. L e t G be a semigroup with a zero a. L e t b be an object not in G. T h e n G U {b} is a semigroup under the multiplication rules

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F I N I T E D I M E N S I O N A L C O N V O L U T I O N A L G E B R A S 7 7

x y = x y as in G for all x, y s G ; x b - b x = b for all x s G ;

b 2 ~ a .

2.3. Theorem. L e t G be a semigroup a n d let e be an object n o t in G. T h e n G U (e} (also written Ge), with the multiplication rules

x y - x y as in G for all x, y s G , e x = x e = x for all x e G e ,

is a semigroup. (G~ is said to be o b t a i n e d from G b y adjoining a unit.) We omit the proof.

2.4. Theorem. L e t G be a semigroup. L e t x 1 be an a r b i t r a r y element of G, a n d let a be an object n o t in G. T h e n G 0 {a}, with the multiplication rules

x y = x y as in G for all x, y s G , x a = x x 1 for all x s G ,

a x = x i x for all x s G , a a ~ x l x l ,

is a semigroup. (This semigroup is said to be obtained from G b y adjoining a repeat element.)

Proof. F o r all u s G U {a}, let u ' = u if u eG, a n d let u ' = x l if u = a . T h e n u v = u ' v' for all u, v e G t3 {a}, a n d consequently u (v w) - u ' (v w)' = u ' (v' w') = (u' v') w'

= ( u v) w

2.5. Theorem. L e t G be a semigroup containing an i d e m p o t e n t element x 1 (x~ = xl).

L e t a be an object n o t in G. T h e n G U {a}, with the multiplication rules x y = x y as in G for all x, y e G ,

x a = x x 1 for all x s G , a x = x l x f o r all x s G , a 2 ~ a,

is a semigroup. (The semigroup G U {a} is said to be o b t a i n e d f r o m G b y i d e m p o t e n t adjunction.)

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78 E D W I N H E W l T T AND H E R B E R T S. ZUCKERMAN

Proof. L e t u ' b e d e f i n e d as in t h e p r o o f of t h e p r e c e d i n g t h e o r e m . T h e n u v - a if u = v = a, a n d u v = u' v' o t h e r w i s e . Since a ' = x I = x 1 x 1 = a ' a ' a n d u' v' ~ G, we h a v e ( u v ) ' = u ' v' for all u, v e G tJ {a}. T h e n u ( v w ) = u ' ( v w ) ' = u ' ( v ' w ' ) = ( u ' v ' ) w ' = ( u v ) ' w'

= ( u v ) w unless u = v = w = a. Since a ( a a ) = (aa)a = a, t h e p r o o f is c o m p l e t e .

T h e r e m a i n d e r of t h e p r e s e n t ~ection is d e v o t e d t o a s t u d y of t h e a l g e b r a of s e m i g r o u p s a n d of t h e s t r u c t u r e of c e r t a i n classes of s e m i g r o u p s . A few of t h e r e s u l t s set f o r t h here a r e t o b e f o u n d in one o r a n o t h e r f o r m in p a p e r s of Clifford [10], P e e l e [30], R e e s [31, 32], S c h w a r z [36, 37, 38], a n d Su~kevi5 [42]. F o r r e a s o n s of n o t a - t i o n a n d s u b s t a n c e , we f i n d i t wise t o give a c o m p l e t e discussion.

2.6. Definition. L e t x b e a n e l e m e n t of a s e m i g r o u p . W e shall s a y t h a t x is of f i n i t e o r d e r s if t h e r e e x i s t 2 i n t e g e r s k > l a n d l > 1 such t h a t xk+l=x l. I t is e a s y t o see t h a t all e l e m e n t s of a f i n i t e s e m i g r o u p a r e of f i n i t e order. W e n o w list s o m e s i m p l e p r o p e r t i e s of e l e m e n t s of f i n i t e order.

2.6.1. I f x is of f i n i t e o r d e r , t h e s e q u e n c e x, x z, x a . . . . c o n t a i n s a t m o s t k + l - 1 d i s t i n c t e l e m e n t s . I f r is t h e s m a l l e s t i n t e g e r such t h a t x T = x s, l<__s<r, we l e t l x = s , k z = r - s . T h e n x € q, p > q , if a n d o n l y if q>_lz a n d p = q + ] k ~ , for s o m e i n t e g e r ~. 9

2.6.2. I f x is of f i n i t e order, t h e n ( x m ) 2 = x m if a n d o n l y if t h e c o n d i t i o n s m=~kx>_lx hold, for s o m e p o s i t i v e i n t e g e r ~. T h i s follows a t once f r o m 2.6.1

2.6.3. I f (x'~)2=x m for s o m e m_> 1, t h e n x is of f i n i t e order.

2.6.4. I f ( x ~ ) Z = x m a n d (xr)2=X r, t h e n x rn= (xm) r = (xr) rn =X r.

2.6.5. I f X is of finite o r d e r a n d l~ = 1, t h e n (x~z) ~ = x k~.

2.6.6. I f G is a s e m i g r o u p all of whose e l e m e n t s are of f i n i t e o r d e r a n d if, f u r t h e r m o r e , G h a s a left u n i t e, a n d G has j u s t one i d e m p o t e n t e l e m e n t , t h e n G is a g r o u p . T h i s follows from 2.6.2, since we t h e n h a v e (xm)2=X m, e 2 = e , a n d h e n c e xm=e. T h e n we h a v e x e - x x m = x m x = e x = x a n d

XX~

t h i s show's t h a t t h e s e m i g r o u p G is a g r o u p . ( I n t h e finite case, t h i s follows e a s i l y f r o m T h e o r e m 39 of S c h w a r z [37].)

2.7. T h e o r e m . L e t G be a f i n i t e or infinite, c o m m u t a t i v e , i d c m p o t c n t semi- g r o u p . T h e n G h a s a c o n c r e t e r e p r e s e n t a t i o n as a s y s t e m of s u b s e t s of t h e e l e m e n t s of G.

s F o r a n e x t e n s i v e u t i l i z a t i o n of t h i s n o t a t i o n , see SCHWAaZ [36].

a A n i n t e r e s t i n g t h e o r e m c o n c e r n i n g c e r t a i n s e m i g r o u p s i n w h i c h

Iz

= 1 for a l l x is t o be f o u n d i n GREEN a n d REES [17].

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F I N I T E D I M E N S I 0 1 q A L C O N V O L U T I O N A L G E B R A S 79 Proof. For each x ~ G, we let M~ = { z x ; z ~ G}, the set of all multiples of x. We have x e M z and x y t M ~ f3 M y , so t h a t all of the sets Mz and all finite intersections of sets Mx are non-void. If M ~ = M y , then x = u y and y = v x for s o m e u, v t G . We thus find x = u y = u y y = u y v x = x v x = v x = y . Furthermore, if z e M x N M u , then z = u x = v y for some u, v e G and we have z = z ~ = u v x y e M ~ . Conversely, if z ~ M ~ , then z = u x y = ( u x ) y = ( u y ) x and z e . M ~ N M ~ . Therefore M ~ = M z N M ~ and the mapping x--->Mz is an isomorphism of G onto the semigroup of sets {Mx}~c, the semigroup operation being set-theoretic intersection.

2.7.1. As a converse to Theorem 2.7, we have: If S is a n y set and G is a set of subsets of S such t h a t the intersection of every pair of subsets belonging to G is in G, then G is a commutative idempotent semigroup under the operation of intersection.

2.7.2. Remark. Theorem 2.7 and its converse provide an interesting sidelight to Stone's representation theorem [41] for Boolean rings. Let A be a Boolean ring, i.e., an associative ring in which x z = x for all x e A. I t is an elementary fact t h a t A is commutative. Thus A, under multiplication alone, forms an idempotent commutative semigroup with a zero; A m a y or m a y not have a unit. I t is easy to show t h a t if A contains more than 2 elements, then the zero is not adjoined: A contains divisors of zero. Stone's theorem asserts t h a t A admits a concrete representation as a ring .,I of subsets of a certain set, the operation a + b of A becoming (A (I B ' ) U ( A ' 0 B) in

~t and a . b in A becoming A N B in I4. Theorem 2.7 shows that if we wish merely to represent the multiplication operation in A, we can use the simple mapping a ~ { z a ; z e A } to obtain a faithful representation of A by sets. However, this mapping shows no tendency at all to preserve the operation + , and we have no hope of ob- taining by this method an elementary proof of Stone's theorem.

2.8. Theorem. Suppose t h a t G is a finite or infinite commutative semigroup, all of whose elements are of finite order. If Ix = 1 for all x s G , then G consists of a set of disjoint groups.

Proof. For each idempotent element a e G , we take Sa = ( x ; x '~ =a}, the set of all x e G such that x m = a for some m. From 2.6.4, we see t h a t the S~ are disjoint and from 2.6.2, we see t h a t every x e G is in some S~. If x, y e S a, then, using 2.6.5 and 2.6.4, we have

(x y)~x~y = (xkx)~, ( y ~ ) k x = x~x yky = a 2 = a

and hence x y e S a . We also have a x = x k z § and x x ~k.~ ~ = x 2 k x = a ; thus a is the unit element of Sa and x 2kx-1 is the inverse of x in S~.

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8 0 E D W I N H E W I T T AND H E R B E R T S. ZUCKERMAN

I f G is a s e m i g r o u p s a t i s f y i n g t h e c o n d i t i o n s of T h e o r e m 2.8, t h e n t h e set H of i d e m p o t e n t e l e m e n t s of G is itself a s e m i g r o u p . E a c h a e H is t h e u n i t of a g r o u p Sa.

I f t h e i d e m p o t e n t s e m i g r o u p H a n d t h e c o r r e s p o n d i n g g r o u p s Sa a r e k n o w n , t h e n G w o u l d be c o m p l e t e l y d e t e r m i n e d if t h e p r o d u c t s of e l e m e n t s of d i f f e r e n t S~ were k n o w n . W e will n o t a t t e m p t to c h a r a c t e r i z e t h e s e s e m i g r o u p s c o m p l e t e l y 9a. H o w e v e r , t h e following r e s u l t s will be useful.

2.9. T h e o r e m . I f G is as in 2.8 a n d x e S ~ , y e S o , t h e n x y e S , b.

Proof. W e h a v e ( ~ y ) k x % - - x k ~ y % ~ a b , a n d t h e r e f o r e x y e S a ~ .

2.10.1. I f x e S a a n d y e S o , t h e n x y , bx, a n d a y all b e l o n g to S~b. T h e r e f o r e we h a v e x y = a b x y = ( b x ) ( a y ) , so t h a t t h e p r o d u c t s x y a r e d e t e r m i n e d once we k n o w t h e p r o d u c t s b x for x e S a a n d a y for y e S b , t o g e t h e r w i t h m u l t i p l i c a t i o n in t h e g r o u p Sab.

2.10.2. If x ~ S , , b2=b, a n d a b = a , t h e n b x = b ( a x ) = a x - x . I n o t h e r words, t h e i d e m p o t e n t b is n o t o n l y t h e u n i t for t h e g r o u p S~, b u t is also a u n i t for S = (J Sa. I t is e a s y t o see t h a t S is a s e m i g r o u p .

a b - a

2.11. W e now c o n s i d e r a f i n i t e or i n f i n i t e c o m m u t a t i v e s e m i g r o u p G all of whose e l e m e n t s a r e of f i n i t e order. L e t us d e n o t e t h e set ( x ; x e G, lx = 1} b y t h e s y m b o l G ~ I f x, y s G ~ t h e n , b y 2.6.5, we h a v e

(xy)k~%:l = (xk~)k~, (y%)% x y = x k ~ y% xy=xkX+l yk~+l = x y .

This i m p l i e s t h a t l ~ y - 1 , in view of 2.6.1. T h e r e f o r e G ~ is a s e m i g r o u p a n d it satis- fies t h e c o n d i t i o n s of T h e o r e m 2.8. I f H d e n o t e s as a b o v e t h e set of i d e m p o t e n t e l e m e n t s of G, t h e n H ~ G ~ . F o r each a s H , we t a k e T ~ = { x ; x s G , x 'n=a for s o m e i n t e g e r m}. F r o m 2.6.4 a n d 2.6.2, we see t h a t t h e sets Ta a r e p a i r w i s e d i s j o i n t a n d t h a t e v e r y x s G lies in s o m e T~. E a c h T~ is a s e m i g r o u p a n d T a N G ~ is a g r o u p : i t is s i m p l y t h e Sa of T h e o r e m 2.8 a p p l i e d to G ~ I f x s T a a n d y sTb, t h e n x m = a , y~ b, a n d c o n s e q u e n t l y (xy)~'~-(x'~)~(y~)'~-a~bm-ab. I t follows t h a t x y ~ T , ~ .

W e t h u s see t h a t Ta is a s e m i g r o u p c o n t a i n i n g t h e g r o u p T , N G ~ I t is of i n t e r e s t t o n o t e t h a t a T ~ - T ~ N G ~ . To p r o v e this, l e t x be a n y e l e m e n t of Ta.

T h e n we h a v e x m - - a a n d h e n c e ( a x ) m - ~ l = a x m ~ l - a a x = a x , so t h a t / ~ = 1 b y 2.6.1.

I n t h e p r e c e d i n g few p a r a g r a p h s , we h a v e d e a l t o n l y w i t h c o m m u t a t i v e semigroups. W e now d r o p t h a t r e s t r i c t i o n b u t a d d a n o t h e r s t r o n g c o n d i t i o n . W e c o n s i d e r t h e effect of i m p o s i n g a o n e - s i d e d c a n c e l l a t i o n law. 1~ Since it is i m m a t e r i a l w h e t h e r

~'~ But see CLIFFORD [10], Theorem 3.

lo Other structural properties of semigroups with a one-sided cancellation law are given in

TAMARI, Bull. de la Soc. Math. de France, 82 (1954), 53-96.

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F I N I T E D I M E N S I O N A L C O N V O L U T I O N A L G E B R A S 81 i t be left- or right-sided, we confine o u r a t t e n t i o n to t h e left c a n c e l l a t i o n l a w : x y = x z i m p h e s y = z, for all x, y, z r G.

2.12. Theorem. I f G is a finite or i n f i n i t e s e m i g r o u p a l l of whose e l e m e n t s are of finite order, a n d if G obeys t h e left c a n c e l l a t i o n law, t h e n G is t h e direct p r o d u c t of a g r o u p J a n d a s e m i g r o u p H such t h a t x y = y for all x, y e l l . 11

Proof. I f x ~ G a n d x k + Z = x ~, / > 1 , t h e n x ~ l x k + ~ = x Z I x , a n d we cancel t h e factor x l-1 to o b t a i n t h e e q u a l i t y x ~ l = x . Therefore / x = l , b y 2.6.1, for all x e G .

N e x t , let a be a n y fixed i d e m p o t e n t e l e m e n t of G a n d let J = ( x ; x ~ G, x a = x } , H = ( x ; x ~ G , x ~ = x ~ . If x , y e J , t h e n x y a = x y a n d x y e J . F u r t h e r m o r e , a x = a a x , so t h a t x - a x a n d a is a (two-sided) u n i t i n J . Also x x k x = x k x + l = x = x a . H e n c e x k x = a . I t follows t h a t x ~ l x = x x 2 ~ x - l = x 2 k z - - . a 2 = a ; hence x has a n i n v e r s e i n J . Accordingly, J is a group. W e n e x t m a k e t h e following o b s e r v a t i o n s .

2.12.1. I f x ~ H a n d y e G , t h e n x x y = x y a n d therefore x y = y . This m a k e s it o b v i o u s t h a t H is a semigroup.

I f x e G , we let u ~ = x a a n d v ~ = x ~ . T h e n U x a = x a a = x a = u x , so t h a t u x e J . W e h a v e v~ e H from 2.6.5. Since a e H, we c a n use 2.12.1 to find t h e following r e l a t i o n : 2.12.2 Ux vx = x a x kx = x k x + l ~ X.

W e n e x t n o t e t h a t if u e J , v e H , a n d w = u v , t h e n u w = w a = u v a = u a = u , b y 2.12.1. Also, vw = w kw = ( u v ) kw = u ~w v b y 2.12.1. Therefore UVw = u ~w+~ v = ( u v ) ~w+~ = u v . W e cancel t h e f a c t o r u a n d o b t a i n Vw = v.

W e h a v e n o w e s t a b l i s h e d t h e existence of a o n e - t o - o n e correspondence

x~(u~,v~) (xeG, u~eJ, v~eH).

I f z ~ ( u z u y , v~ v~), t h e n , i n view of 2.12.2 a n d 2.12.1, we h a v e z = u x u~ vx vy - ux uy v~

a n d x y = Ux v ~ u ~ v y = u ~ u y v ~ . I t follows t h a t z = x y , a n d hence t h e o n e - t o - o n e correspondence j u s t established is a n i s o m o r p h i s m of G o n t o t h e direct p r o d u c t J X H.

This completes t h e p r e s e n t proof.

2.12.3. Note. According to a t h e o r e m of Clifford l~ e v e r y s e m i g r o u p G i n which there exists a left u n i t e a n d i n which, for all x e G , a n x' exists w i t h x x ' = e , is t h e direct p r o d u c t of a g r o u p a n d a semigroup i n which x y = y i d e n t i c a l l y . C o m b i n i n g this r e s u l t with T h e o r e m 2.12, we h a v e t h e following theorem. L e t G b e

lo~ A n n . o/ Math., 2 Ser., 34 (1933), 865-871.

11 This theorem, for finite semigroups G, follows immediately from results of SUw [42].

His methods are not, however, applicable in the general ease treated here.

6 - 5 4 3 8 0 9 . Acta Mathematica. 93. Imprlm~ lo 10 mai 1955.

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a s e m i g r o u p in which e v e r y e l e m e n t h a s f i n i t e order. T h e n G o b e y s t h e left cancella- t i o n law if a n d o n l y if G h a s a left u n i t e a n d r i g h t i n v e r s e s r e l a t i v e to e.

2.12.4. Note. T h e o r e m 2.12 fails for s e m i g r o u p s c o n t a i n i n g e l e m e n t s of i n f i n i t e order, as t h e n o n - n e g a t i v e i n t e g e r s u n d e r a d d i t i o n show.

W e t u r n n o w t o t h e case of a n i d e m p o t e n t s e m i g r o u p t h a t is n o t n e c e s s a r i l y c o m m u t a t i v e . 12

2.13. I f G is a f i n i t e or i n f i n i t e i d e m p o t e n t s e m i g r o u p , t h e n G consists of a set of d i s j o i n t s e m i g r o u p s , as follows. W e t a k e S~ = { x ; x e G, a x a = a, x a x = x } , for e a c h a e G . Since a a a = a , we h a v e a e S ~ so t h a t no Sa is v o i d a n d e v e r y x e G lies in s o m e Sa.

I f x, yeS~, we h a v e x y a x - ( x a x ) y a x = x ( a y a ) x y a x = x a ( y a x ) ( y a x ) = x a ( y a x ) =

= x ( a y a ) x = x a x = x , a n d h e n c e x y x = x y ( x y a x ) = ( x y ) ( x y ) a x = x y a x = x . T h e n we also h a v e y x y = y a n d t h e r e f o r e x e S y , yeS~. T h i s shows t h a t x e S ~ i m p l i e s S x = S ~ a n d t h a t t h e sets S~ a r e p a i r w i s e d i s j o i n t . Also if x, y e S a , we now h a v e x ( x y ) x =

= x y x = x a n d ( x y ) x ( x y ) = x y x y - x y , so that x y e S ~ = S ~ . I t follows t h a t S~ is a s e m i g r o u p .

2.14. T h e o r e m . L e t G, S~, a n d S~ b e as in 2.13. I f XeSa a n d yeS~, t h e n x y e S~b a n d S~a = S~o.

Proof. I f xeS~, t h e n ( b a ) ( x b ) ( b a ) = b a x b a = b a x b ( a x a ) - ( b a x ) ( b a x ) a = ( b a x ) a =

= b(axa) = ba. F u r t h e r m o r e , (xb)(ba)(xb) = x b a x b = ( x a x ) b a x b = x ( a x b ) ( a x b ) =

= x ( a x b) - ( x a x ) b = x b. T h e r e f o r e x b e Sb~. T h i s i m p l i e s t h a t a b e S0~ ; since we h a v e abeS~b, i t follows t h a t S~a--S~0.

I f x e S , a n d y e S b , we h a v e b e S 0 = S y a n d t h e n , f r o m w h a t we j u s t p r o v e d a b o v e , we see t h a t x y e S , ~ a n d b a e S ~ . B u t t h i s i m p l i e s t h a t xyeSy~=So~=S~b.

This c o m p l e t e s t h e p r e s e n t proof.

W e n o t e also t h a t if H is t h e set of d i s t i n c t S , , t h e n H is a c o m m u t a t i v e , i d e m p o t e n t s e m i g r o u p u n d e r t h e o p e r a t i o n Sa So = So0.

2.15. Theorem. E a c h S~ of 2.13 is i s o m o r p h i c w i t h a s e m i g r o u p of p a i r s (y, z) ill which t h e s e m i g r o u p o p e r a t i o n is d e f i n e d b y (Yl, zl)(Y2, z2)--(YD z.).

Proof. W e t a k e ,~"a = (xa; xeSa}, "' Sa = { a x x e S , } . T h e n S a c S a ' a n d S~ c S ~ . "

Also ye,~,"~ i m p l i e s t h a t a y = a , y a = y ; s i m i l a r l y , z e S ' j i m p l i e s t h a t a z = z , z a = a . I f xeS~, t h e n y = x a e S ' a a n d z = a x e S a ' . Also y z = ( x a ) ( a x ) = x a x = x .

12 In com}ection with this topic, see also MCLEAN" [27] and CLIFFORD [10]

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FINITE DIMENSIONAL CONVOLUTION ALGEBRAS 83

~s ~tl

I f y s a a n d z s a , t h e n w e h a v e x = y z s S a a n d x a = y z a = y a = y , a x = a y z =

= a z : z .

W e h a v e t h u s e s t a b l i s h e d t h e e x i s t e n c e of a o n e - t o - o n e c o r r e s p o n d e n c e x+-*(xa, a x) b e t w e e n S~ a n d t h e .set Ta of p a i r s (y,z) f o r y sSa, z sS'j. I f x l ~ ( y 1,z,) a n d x 2 ~ (y~, z2), t h e n w e h a v e x 1 x 2 = Yl zl Y2 z2 = (Yl a) z I (Y2 a) z 2 : Yl (a z 1 Y2 a) z 2 = YI a z2 - Yl z2"

T h e c o r r e s p o n d e n c e b e c o m e s a n i s o m o r p h i s m w h e n w e d e f i n e (Yl, zl)(Y2, z 2 ) a s (Yl, z2)- T h i s c o m p l e t e s t h e p r o o f .

I t is of i n t e r e s t t o o b s e r v e t h a t f o r xl, x2~Sa, t h e e q u a l i t y xlx2=x,zx 1 o b t a i n s if a n d o n l y if x I - x 2 .

T h e s e m i g r o u p s S~ a r e c o m p l e t e l y d e t e r m i n e d b y t h e s e t s S'~ a n d S ' j . F o r e x a m p l e , if S~ is f i n i t e , w e h a v e x ~ ( i , ] ) , l<_i<_Ic, l<_]<_l, w h e r e k is t h e n u m b e r of e l e m e n t s in S " a n d l t h e n u m b e r i n S " .

I f ysS'a a n d z s S ' j , t h e n y+-~(ya, a y ) = ( y , a ) a n d z~-~(za, a z ) = ( a , z ) . W e also h a v e Yl Y~ = Yl if Yl, Y2 s S'~ a n d z lz~ = z 2 if z 1, z 2 s S ' j . F u r t h e r m o r e , if y s S~ a n d x s S~,

t ~tt

t h e n x y o ( x a , a x ) ( y , a ) = ( x a , a), so t h a t x y s S ~ . S i m i l a r l y , we f i n d t h a t z x s a if z ~ S ' j a n d x e S~. W e will s a y t h a t S~ is a l e f t i d e a l of S~ a n d S~' is a r i g h t i d e a l of S~.

2.16. D e f i n i t i o n . A n o n - v o i d s u b s e t I of a s e m i g r o u p G is s a i d t o b e a l e f t i d e a l of G if x y s I f o r all x s G a n d y s I . R i g h t a n d 2 - s i d e d i d e a l s a r e d e f i n e d s i m i l a r l y . 13

w 3. Representations of semigroups

W e u s e t h e t e r m representation ill c o n n e c t i o n w i t h a s e m i g r o u p G t o m e a n a h o m o m o r p h i s m of G i n t o t h e m u l t i p l i c a t i v e s e m i g r o u p ~ f o r s o m e n>_ 1. W e b e g i n w i t h a s t u d y of 1 - d i m e n s i o n a l r e p r e s e n t a t i o n s .

3.1. D e f i n i t i o n . L e t G be a s e m i g r o u p . A c o m p l e x f u n c t i o n Z d e f i n e d o n G is s a i d t o b e a s e m i c h a r a c t e r of G if Z (x) :~ 0 for s o m e x s G a n d Z (xy) = Z (x) Z (Y) f o r all x, y s G . 1~

W e l i s t a f e w s i m p l e p r o p e r t i e s of s e m i c h a r a c t e r s . P r o o f s a r e l e f t t o t h e r e a d e r . 3.1.1. I f x is a n e l e m e n t of f i n i t e o r d e r b e l o n g i n g t o a s e m i g r o u p G a n d if Z is a s e m i c h a r a c t e r of G, t h e n Z ( x ) is e i t h e r 0 o r a r o o t of u n i t y .

3.1.2. I f x is a n i d e m p o t e n t e l e m e n t a n d Z is a s e m i c h a r a c t e r , t h e n Z ( x ) - 0 o r 1.

3.1.3. I f G c o n t a i n s a u n i t , e, a n d if Z is a s e m i c h a r a c t e r of G, t h e n z ( e ) - 1 . 13 This widely used concept goes back at least to SUSKEVIC.

it S. SCHWArtZ [38] has used a slightly different definition of semicharacter and has obtained a number of interesting results paralleling ours. We are also indebted to Dr. SCHWA~Z for personal conversations on this topic.

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3.1.4. I f G c o n t a i n s a zero, z, a n d if Z is a s e m i c h a r a c t e r of G n o t i d e n t i c a l l y 1, t h e n Z (z) = 0.

3.1.5. A s e m i c h a r a c t e r of a g r o u p is a c h a r a c t e r in t h e u s u a l sense.

I t is e s s e n t i a l for o u r p u r p o s e s t h a t a s e m i c h a r a c t e r be a l l o w e d to a s s u m e t h e v a l u e 0. I n f a c t :

3.1.6. L e t G be a f i n i t e s e m i g r o u p such t h a t for e v e r y p a i r of d i s t i n c t e l e m e n t s x, y e G , t h e r e e x i s t s a s e m i c h a r a c t e r Z v a n i s h i n g n o w h e r e such t h a t g ( x ) : # Z ( y ).

T h e n G is a c o m m u t a t i v e g r o u p .

3.2.1. F o r t h e m o m e n t , we c o n s i d e r a f i n i t e c o m m u t a t i v e s e m i g r o u p G a n d s u p - pose t h a t g is a s e m i c h a r a c t e r of G. W e will use t h e n o t a t i o n a n d r e s u l t s of 2.11.

F o r some x ~ G , we h a v e Z ( x ) # 0 " T h e n 2.6.2 shows t h a t x ( a ) # 0 for s o m e a e H , a n d h e n c e z ( a ) = 1, b y 3.1.2. W e t a k e % = 1-[ a. I t is clear t h a t Z ( a o ) = 1 a n d

aeH, x ( a ) = l

t h a t for a e H , we h a v e z ( a ) = l if a o a = a o a n d z ( a ) = O if a o a # % . F o r all x e G , we h a v e g ( x ) = z ( a o ) ) ~ ( x ) = y , ( a o X ) . I f x e T , ~ a n d a o a = a o , t h e n a o x e T a . , ~ = T a ' a n d a o x = a o ( a o x ) e a o T , ~ o = T a . N G ~ . I f x e T , ~ a n d a o a : # a o, t h e n x m = a for some m a n d h e n c e Z(x) m = Z ( a ) = 0 , which i m p l i e s t h a t : ~ ( x ) = 0 . N o w T ~ . N G ~ is a g r o u p a n d g ( a 0 ) # 0 ; hence Z, w i t h i t s d o m a i n r e s t r i c t e d t o t h e g r o u p T~, N G ~ is a c h a r a c t e r of t h a t g r o u p . W e h a v e t h e r e f o r e p r o v e d t h e following t h e o r e m .

3.2. T h e o r e m . I f G is a f i n i t e c o m m u t a t i v e s e m i g r o u p a n d Z is a s e m i c h a r a c t e r of G, t h e n t h e r e is a n e l e m e n t % e l l a n d a c h a r a c t e r :~a. 15 of t h e g r o u p T~. N G ~ s u c h t h a t

0 if a 0 a # a 0 for t h e e l e m e n t a such t h a t x ~ T ~ , Z ( x ) = g a . ( % x ) if a o a = a o for t h e e l e m e n t a such t h a t x s T ~ .

3.2.2. W e p o i n t o u t t h a t T h e o r e m 3.2 r e m a i n s v a l i d for c o m m u t a t i v e s e m i g r o u p s G in which all e l e m e n t s h a v e f i n i t e o r d e r a n d in w h i c h H is finite.

3.3. T h e o r e m . I f G is a f i n i t e o r i n f i n i t e c o m m u t a t i v e s e m i g r o u p , all of whose e l e m e n t s h a v e f i n i t e order, if a 0 e H , a n d if Xa. is a c h a r a c t e r of t h e g r o u p Ta. N G ~ t h e n t h e f u n c t i o n :~, d e f i n e d b y t h e r e l a t i o n s

X (x) = {

^{0 if}a o a # a o for t h e e l e m e n t a such t h a t x e T a , Za.(aox ) if a o a = a o for t h e e l e m e n t a s u c h t h a t x e T a , is a s e m i c h a r a c t e r of G.

is F o r t y p o g r a p h i c a l r e a s o n s , w e v i o l a t e h e r e t h e c o n v e n t i o n o f f o o t n o t e 4.

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F I N I T E D I M E N S I O N A L C O N V O L U T I O N A L G E B R A S 85 Proof. Since Z(ao)~=0, we h a v e o n l y t o p r o v e 24(xy)=24(x)24(y). I f x e T a a n d y e T b , t h e n x y e T a b . I f a o a b = a o, t h e n a o a = a o a b a - - a o a b = a o a n d a o b = a o a b b =

= a o a b = a 0. I f a 0 a = a o b = a 0 , t h e n a o a b = a 0 a a o b = a o. T h e r e f o r e we h a v e 24(xy) 4=0 if a n d o n l y if Z (x) 4= 0 a n d 24 (y) ~: 0. I f 24 (x) :~ 0 a n d 24 (y) 4 0, we t h e n h a v e 24 (x) 24 (y) =

= 24~0 (a o x) 24~. (a o y) = z~, (ao x a o y) = Za, (ao x y) = 24 (x y).

W e h a v e also t h e following s i m p l e c o n s e q u e n c e s of T h e o r e m s 3.2 a n d 3.3.

3.3.1. Corollary. L e t G b e a f i n i t e c o m m u t a t i v e s e m i g r o u p . T h e n t h e s e m i c h a r - a c t e r s of G f o r m a l i n e a r l y i n d e p e n d e n t set of functions.

P r o o f . I f 241 . . . 24~ a r e s e m i c h a r a c t e r s of G, if ~1 . . . r 1 6 2 a n d ~ ccs g j = 0 ,

j - 1

t h e n consider a n y i d e m p o t e n t a s G a n d t h e g r o u p Ta fl G ~ On t h i s g r o u p , e v e r y s e m i c h a r a c t e r 24j is e i t h e r i d e n t i c a l l y 0 or is a c h a r a c t e r . Since c h a r a c t e r s of a f i n i t e g r o u p are l i n e a r l y i n d e p e n d e n t , we see t h a t ~ j = 0 for all ?" such t h a t z j ( a ) 4 0 . Since a is a r b i t r a r y , i t follows t h a t all a t = 0 (~= 1, 2 . . . m).

3.3.2. Corollary. L e t G b e a f i n i t e c o m m u t a t i v e s e m i g r o u p a d m i t t i n g m d i s t i n c t s e m i c h a r a c t e r s . T h e n m_< o (G).

3.3.3. Note. Corollaries 3.3.1 a n d 3.3.2 r e m a i n v a l i d for n o n c o m m u t a t i v e f i n i t e s e m i g r o u p s G : one can show t h i s b y m a p p i n g G h o m o m o r p h i c a l l y o n t o a c o m m u t a t i v e s e m i g r o u p H a d m i t t i n g j u s t t h e s a m e s e m i c h a r a c t e r s as G. W e o m i t t h e d e t a i l s .

3.4. T h e o r e m . L e t G b e a s e m i g r o u p all of whose e l e m e n t s a r e of f i n i t e o r d e r a n d h a v i n g t h e p r o p e r t y t h a t for all x, y e G such t h a t x=#y, t h e r e is a s e m i c h a r a c t e r Z such t h a t Z (x)~=X (Y). T h e n G is c o m m u t a t i v e a n d Ix = 1 for all x e G.

Proof. F o r all s e m i c h a r a c t e r s Z, we h a v e Z (x y) = Z (x) Z (Y) = Z (Y) g (x) = X ( y x ) . I t follows t h a t x y = y x . Also X (x) k~: +ix = Z ( xkx ~Zx) = 24 (xl'~) = X (x) ix a n d h e n c e Z (x) ~x+l =

=24(x) for all 24: t h u s we h a v e x ~ x + l = x , a n d b y 2.6.1, l x = l .

3.5. T h e o r e m . L e t G b e a f i n i t e or i n f i n i t e c o m m u t a t i v e s e m i g r o u p all of whose e l e m e n t s a r e of f i n i t e o r d e r a n d for w h i c h l~ = 1 for all x e G. T h e n , for e v e r y x, y e G such t h a t x : # y , t h e r e is a s e m i c h a r a c t e r 24 such t h a t 24(x)=#24(y).

Proof. I f 24 (x)=24 (y) for all of t h e s e m i c h a r a c t e r s 24 d e s c r i b e d in T h e o r e m 3.3, we first t a k e a o = x ~x a n d o b t a i n 24 (x) = 24a~ (% x) = 24a0 (x kxx) = 24a~ (x). Since 24a~ (x) 4= 0, we h a v e 24 (y) = 24 (x) 4 0 a n d hence a o y~v = ao, 24 (y) = 24a, (x k3: Y). T h u s we h a v e Za~ (x) =

= Zao (xkxy) ; b u t Za, c a n b e a n y c h a r a c t e r of t h e c o m m u t a t i v e g r o u p T~, fl G ~ (which in t h i s case is j u s t Tao), so we h a v e x = x k x y . This follows f r o m t h e w e l l - k n o w n t h e o r e m

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t h a t d i s t i n c t e l e m e n t s of a n y c o m m u t a t i v e g r o u p can be s e p a r a t e d b y c h a r a c t e r s . See for e x a m p l e W e i l [48], p. 99. W e also h a v e xk::ykY=x kz since %y~Y=%. Since x a n d y are q u i t e i n t e r c h a n g e a b l e in t h i s a r g u m e n t , we also h a v e y = y ~ y x a n d yky x~x = y~y. T h e r e f o r e x kx = x ~x y ~ = y~y a n d x = x k~ y = yk~ y = y. T h i s c o m p l e t e s t h e proof.

3.6.1. R e t u r n i n g t o t h e case of a finite c o m m u t a t i v e s e m i g r o u p G, we see b y T h e o r e m 3.2 t h a t all s e m i c h a r a c t e r s of G a r e g i v e n b y T h e o r e m 3.3. E a c h semi- c h a r a c t e r of G, is, of course, a s e m i c h a r a c t e r of G ~ I n t h e o t h e r d i r e c t i o n , i t follows f r o m T h e o r e m 3.3 t h a t each s e m i c h a r a c t e r of G ~ can b e e x t e n d e d t o b e a s e m i c h a r - a c t e r of G in one a n d o n l y one w a y . I n fact, if x ~ Ta a n d x non e G ~ t h e n a x e G ~ a n d xm=a for s o m e m > _ l . T h e r e f o r e Z ( x ) = z ( a ) z ( x ) = z ( a x ) if z ( a ) = l a n d z ( x ) = 0 if Z (a) - 0, or m o r e s h o r t l y , Z (x) = Z (a) :~ (a x).

3.6.2. W e also n o t i c e t h a t if G is a f i n i t e c o m m u t a t i v e s e m i g r o u p , t h e n t h e n u m b e r of d i s t i n c t s e m i c h a r a c t e r s of G is j u s t t h e n u m b e r of e l e m e n t s of G ~ T h i s follows f r o m T h e o r e m s 3.2 a n d 3.3, since t h e n u m b e r of c h a r a c t e r s of a finite c o m - m u t a t i v e g r o u p is j u s t t h e n u m b e r of its e l e m e n t s .

3.7. L e t G be a s e m i g r o u p . L e t G d e n o t e t h e set of all s e m i c h a r a c t e r s of G.

F o r Z, ~PeG, t h e p r o d u c t Z~0 is t h e f u n c t i o n on G such t h a t Z ~ ( x ) = z ( x ) ~ f ( x ) for all x e G .

W e list a n u m b e r of s i m p l e results, l e a v i n g t h e proofs t o t h e r e a d e r .

3.7.1. T h e s e m i c h a r a c t e r s of a s e m i g r o u p e i t h e r f o r m a s e m i g r o u p b y t h e m s e l v e s or t h e y f o r m a s e m i g r o u p if a n a d d i t i o n a l e l e m e n t 0 is s u p p l i e d . See s e m i g r o u p s 1 a n d 5, A p p e n d i x 2.

3.7.2. I f G is a s e m i g r o u p , t h e n 0 h a s a u n i t .

3.7.3. I f G is a s e m i g r o u p w i t h a u n i t , t h e n 0 is a s e m i g r o u p .

3.7.4. I f G a n d G a r e s e m i g r o u p s , t h e n , in t h e n o t a t i o n of T h e o r e m s 2.2 a n d 2.3, 0 ~ - (G)z a n d ( ~ = ((~)~.

3.7.5. I f Z is a s e m i c h a r a c t e r of a s e m i g r o u p G a n d if e ( x ) - 1 for Z ( x ) + 0 a n d e ( x ) - 0 for Z ( x ) = 0 , t h e n e is a s e m i c h a r a c t e r of G.

3.7.6. L e t G b e a s e m i g r o u p a n d l e t Z be a s e m i c h a r a c t e r of G. L e t ~p(x)

= [Z (x)]-I for Z (x) ~ 0 a n d ~f (x) = 0 for Z (x) = 0. T h e n ~f (x) is also a s e m i c h a r a c t e r of G.

3.7.7. I f Z is a s e m i c h a r a c t e r , t h e n so is 2..

3.8. L e t H be a f i n i t e c o m m u t a t i v e i d e m p o t e n t s e m i g r o u p . I f a e H , t h e n we s a y t h a t a is a p r i m e e l e m e n t of H if t h e e q u a l i t y a = b c ( b , c e H ) i m p l i e s b = c = a .

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F I N I T E D I M E N S I O N A L C O N V O L U T I O N A L G E B R A S 87 3.8.1.

i m p l y b = a.

3.9. T h e o r e m . one p r i m e e l e m e n t .

A n e l e m e n t a of H is a p r i m e e l e m e n t if a n d o n l y if a = a b a n d b e l l

E v e r y f i n i t e c o m m u t a t i v e i d e m p o t e n t s e m i g r o u p c o n t a i n s a t l e a s t

Proof. L e t a 1 be a n y e l e m e n t of H. I f aj is n o t a p r i m e e l e m e n t , t h e n t h e r e is a n e l e m e n t a s ~ H such ~hat a l = a l a 2 a n d a 2 ~ a 1. R e p e a t i n g t h i s step, we o b t a i n a sequence al, a 2 . . . am such t h a t a ~ = a ~ a i ~ l a n d a~+l~=a~ ( l < i _ < m - 1 ) . I f h<_]<_m, we h a v e a h = a h a h + x a h ~ 2 . . . a j a n d h e n c e a h a j = a ~ . I f h < ] < _ m a n d a h = a j , t h e n ah = ah ah+~ = ai ah ~1 ^~ a h e l , a n d t h i s is a c o n t r a d i c t i o n . T h e r e f o r e t h e e l e m e n t s a~ a r e all d i s t i n c t , a n d , H being finite, t h e sequence a~, a 2 .. . . . am will e v e n t u a l l y e n d w i t h a p r i m e e l e m e n t a~.

3.9.1. I t is now clear t h a t for e v e r y e l e m e n t a~ ~ H , t h e r e is a p r i m e e l e m e n t a~ of H such t h a t a l = a l a m .

3.9.2. I f ' H h a s j u s t one p r i m e e l e m e n t p, t h e n a = a p for all a e H a n d hence p is a u n i t of H . I t is e a s y to see t h a t if H h a s a u n i t e, t h e n H has j u s t t h e one p r i m e e l e m e n t e.

3.10. T h e o r e m . L e t G b e a c o m m u t a t i v e s e m i g r o u p all of whose e l e m e n t s a r e of f i n i t e o r d e r a n d l e t H be t h e s e m i g r o u p of i d e m p o t e n t e l e m e n t s of G. I f H is finite, t h e n G is a s e m i g r o u p if a n d o n l y if H has a unit.

P r o o f . T h e e l e m e n t s of G a r e t h e s e m i c h a r a c t e r s of G, as c o n s t r u c t e d in T h e o r e m 3.3. L e t X~ b e a n y of t h e s e m i c h a r a c t e r s d e t e r m i n e d b y a o = b , Z2 a s e m i c h a r a c t e r d e t e r m i n e d b y a 0 - c , where b a n d c are e l e m e n t s of H . T h e n X1Z2 fails t o b e a s e m i c h a r a c t e r if a n d o n l y if Z1 (x)X2 ( x ) = 0 for all x e G ; t h a t is, if a n d o n l y if b a 4 b or c a m c for e v e r y a ~ H . I f H h a s a u n i t e, t h e n b e = b a n d c e = c a n d Z1X2 is a s e m i c h a r a c t e r of G. I f H does n o t h a v e a u n i t , t h e n 3.9 a n d 3.9.2 i m p l y t h a t H h a s a t l e a s t t w o p r i m e e l e m e n t s . I f b a n d c a r e d i s t i n c t p r i m e e l e m e n t s of H , t h e n b a r b if a : # b a n d c a 4 c if a = b , a n d t h e r e f o r e t h e p r o d u c t X1X2 is i d e n t i c a l l y 0 a n d is n o t a s e m i c h a r a c t e r of G

3.11. T h e o r e m . L e t G b e a c o m m u t a t i v e s e m i g r o u p all of whose e l e m e n t s a r e of f i n i t e order, a n d l e t H b e a s u s u a l t h e s e m i g r o u p of i d e m p o t e n t e l e m e n t s of G. I f H is finite, t h e n 16 ~ G ~

la The symbol " ~ " is taken to mean the existence of a one.to-one correspondence ~ such that T (x y) = v (x) v (y) whenever either side exists.

FINITE DIMENSIONAL CONVOLUTION ALGEBRAS