THE EMBEDDING OF A TOPOLOGICAL SEMIGROUP IN A TOPOLOGICAL GROUP AND ITS GENERALIZATIONS, AND AN ERGODIC THEOREM FOR A NON-COMMUTATIVE SEMIGROUP OF LINEAR OPERATORS

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8013688 t

t [

PECK, JOHN EDWARD L. THE EMBEDDING OF A TOPOLOGICAL SEMIGROUP IN A TOPOLOGICAL GROUP AND ITS GENERALIZATIONS, AND AN ERGODIC THEOREM FOR A NON-COMMUTATIVE SEMIGROUP OF LINEAR OPERATORS Yale University

University M icrofilm s International

PH.D.

300 N. Zeeb Road, Ann Arbor, MI 48106

1950

18 Bedford Row, London WC1R 4EJ, England

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! i

TEE EMBEDDING 0? A TOPOLOGICAL SEMIGROUP IN A TOPOLOC-ICAL GROUP AMD ITS GENERALIZATIONS,

i

I i !

and

(

| ! I

All ERGODIC THEOREM FOR A 11Oil-COMMUTATIVE SEMIGROUP 0? LI11EAR OPERATORS.

$ J (

|

John S.L.Peck

A Dissertation presented

to the Faculty of the

. Graduate School of Yale University in Candidacy for the Degree of Doctor of Philosophy

1950

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SUHE&RY

This dissertation consists of two chapters which are not related, except that there is a theorem of topological algebra which is common to both. The first chapter deals with generalizations of the problem of embedding a topological semigroup in a topological group.

First, both the commutative and the associative laws

are replaced by the weaker law of alternation.

We are

therefore concerned with alternation topological groupoids. Then while retaining the associative law, the commutative law is replaced by a law which, roughly speaking, postulates the existence of common multiples on one side.

Some results

are also obtained for compact semigroups. Xn the second chapter it is proved that a totally bounded semigroup of linear bounded operators, acting on a Banach space, and satisfying certain generalized commutativity postulates, is an ergodic semigroup and every element of the Banach space is ergodic with respect to this semigroup.

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The author -cashes to thanh Professor Dinar Hille, and other members of the Department of Mathematics, uho by their encouragement and advice, have contributed to the preparation of this dissertation.

i

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TABLE OP COHTEHTS

Chapter 1.

The embedding of a topological semigroup in a topological group and its generalizations page

Chapter-2.

1

An ergodic theorem for a non-commutative semigroup of linear operators.

Bibliography

..

•.

•*

••

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page

40

page

53

CHAPTER 1 .

THE EMBEDDING OP A TOPOLOGICAL SEMIGROUP II'I A 6 E

TOPOLOGICAL GROUP AHD ITS GSHSRALIZATIOUS.

K I ? I; |

Introduction.

We shall use the term groupoid to

denote a set of elements S such that, to any two elements a t-

and b in S there is defined a unique product ab in S.

An

I I

element a in a groupoid S is a left (right) canceller if for

I

any pair of elements x and y in S, the equation (xa*xy)

implies that

x — y.

a x = ay

An element is a canceller

if it is both a left and a right canceller.

An element a

in a groupoid S is left (right) proper if to any element b in S there is a uniquely defined element c in S such that ac=-b

( ca = b ).

and right proper.

An element is proper if it is both left A groupoid is associative if the relation

(ab)c^a(bc) holds between any three elements a,b,c. associative groupoid is called a semigroup.

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1

An

A groupoid

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in which, all elements are cancellers is called a cancellation groupoid, and an associative cancellation groupoid is called a cancellation semigroup*

A groupoid in which every element

is proper is a quasigroup and an associative quasigroup is a group in the usual sense.

For quick reference these notions

may be conveniently tabulated thus:-

an operation

groupoid all elements cancellers

cancellation groupoid

all elements proper

quasigroup

an associative operation semigroup cancellation semigroup group

Any one of the above will be referred to as a system. Unless it is stated otherwise, we shall use the term topology to mean a Hausdorff or separated topology Jl: p32j . We shall introduce topologies into .these systems in the fol­ lowing manner.

A system-with-topology is a system which

is also a topological space, but this topology is not neces­ sarily related to any operation between elements of the system. A topological groupoid S is both a groupoid and a topological space, in which the correspondence

(a,b)->ab

is a contin­

uous function from the product space SxS into S.

A topological

quasigroup is both a quasigroup and a topological space in

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which any one of the three elements a,b,c & IK

i

£• U

related by

ab = c

is a continuous function of the ordered pair formed by the other two in SxS.

A touological semigroup is an associative

topological groupoid, while a topological group is an associa­ tive topological quasigroup.

This coincides with the ortho­

dox definition of a topological group. It is well known that a commutative cancellation semi­ group may be embedded in a group.

In fact, [2: p27] a com­

mutative semigroup may be embedded in another semigroup in which the new set of cancellers forms a group containing the original set of cancellers. £ K: £ s tv r fvi-'

I

|

Malcev £53 has shown that a

non-commutative cancellation semigroup may not, in general, be embedded in a group, and he has also discovered complicated necessary and sufficient conditions [4j].

Ore L 5 3 has given

a construction of the embedding process in the case where the cancellation semigroup, has common multiples on one side, and a similar construction is made by Dubreil [6: pl42j in the case of non-commutative semigroups, where not all elements are cancellers and with a weaker common multiple property. She lander [73 has proved a similar theorem without assuming the associativity.

He defines an alternation groupoid to

be a groupoid where the relation

(ab)(cd)= (ac)(bd)

satisfied between any four elements.

is

We observe that this

is a weakening of both the associative and the commutative laws.

We say that an element a is a '"'canceller if every — o -

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power [7] of a is a canceller.

Sholander proves that an

alternation groupoid may be embedded in another alternation groupoid, which is such that the new set of ""'cancellers is a quasigroup, which contains the original set of ""'cancellers. In particular then any alternation cancellation groupoid may |

I I | f I

be embedded in a quasigroun. In the well known case of embedding a commutative cancellation semigroup S in a group G, we recall that the first step is to set up an equivalence relation in the product semigroup SxS defined by

(a^,ag) ~ (b^,bg) if an only if a-j_-fbg =. ag + b^.

I f|

Using square brackets to denote the equivalence class generated by a pair, we define the operation between equivalence

|

classes by

Eal,a2l

Eal + ^ ^ a g + bgj.

® ie se^

r

|

equivalence classes with this operation forms the group G

|

v/hile the correspondence

|

of the semigroup in the group.

| I: | tr' V §:i !-■ F |

s->[a,a+s]

defines an embedding

If S is also a topological semigroup then there is a natural tocology in the product semigroup SxS, while the “* ^ equivalence relation introduces a topology,- in a natural way, in the set of equivalence classes G, viz. the finest

£•

I

topology for v/hich the correspondence

| tt : r.-‘

SxS to G is continuous.

p

|

topology.

j[

a topological group.

(a,b) —

Ea»'°3

from

Ihis topology is not necessarily

separated, but in a sense we have that G is a group-withIt is therefore natural to ask whether G is Since the topology in SxS is sym-

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metric and since the inverse of

in G is [a^a-jl, it

is clear that the inverse operation in G is continuous.

For

iE;

G to be a topological semigroup it is sufficient that the

£

equivalence relation be open.

fe

what restrictions are required of the original topological

f e K1

It is not immediately obvious

i:V

semigroup.

We propose to discuss this problem, but in a

more general way as follows Section 2 is concerned with some preliminary topological theorems, some of which are common to all the later sections. In section 3 we replace both the associative and the commutative laws by the weaker postulate of alternation and base our topological investigations on the work of Sholander M . In section 4 we keep the associative law, but weaken the commutativity and use the algebraic results of Dubreil [6]. These two sections therefore constitute generalizations in two different directions. In section 5 we show that in the case of a compact topological cancellation semigroup, there is no problem since it is already a topological group. Since we shall quote the work of Dubreil and other French writers, it is well to observe that their demigroup is our semigroup> and their semigroup our cancellation semigroup.

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2.

Topological Groupoids We shall say that a topo­

logical space S is embedded in

a topological space S ’, if

S is homeomorphic with a subset of S* using the relative iX

i> I;

% >':.ftt,ts)lt£ T} - 9 -

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£

2 .6

from S into TxS is open in (2)

TxS.

every element of I is

relative to T, if and only

a continuous left canceller,

if the

mapping s — ^£(t,ts)lt £ T j

from S onto E is open in E, and I?is a subset of the left cancellers. Proof:

(1)

Let every element of 2? be a continuous left

divisor relative to T, and let A be an open set in S. B be its image under the mapping considered. in B.

Let

Let (t,ta) be

Then v/e may find E(t)xE(ta) in TxS such that if

(tVs*) is in E(t)xE(ta) then there is an a.' in A such that s ^ t ’a*.

This proves that ( t S s ’) is in B and that B is

open in TxS.

Conversely let the mapping be open and let B

be the open image of the given open set N(a) in S.

For

every t in T ve have that (t,ts) is in B and T/e may find E(t)3E25"(ta)£ B.

This means that if (t*,s*) is in E(t)xE(ta),

then there is an a* in E(a) such that s ’ - t * a ’, and this proves that t is a continuous left divisor relative to T. (2)

We observe that the expression (t,t

for an element of E is not unique in s unless t is a left canceller.

If every element of T is a continuous left can­

celler relative to T and if A is an open set in S and B its image in K, then let (t,ta) be in B.

Then we may find

E(t)xE(ta) in TxS which is such that if (t’jt’a 1) is in E(t)xE(ta)aE in E.

then a* is in A.

This proves that B is open

Conversely, let a,E(a) in S and t in T be given,

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I

where T Is a set of left cancellers and the image of 3S(a)

|

is open in K.

Since (t,ta) lies in the image of K(a) we

may find N(t)xS(ta) in TxS such that if (t’jt’a 1) lies in K(t)xN(ta)nK, then (t'^t’a 1) is an image of an element of N(a).

But the a ’ in this expression is unique, so that

a 1 is in U(a), which proves that t is a continuous left can­ celler relative to T. Corollary 2.7 .

If every element of T is a continuous

left divisor relative to T, then for any t in T and any open set A of S, tA is open in S. Proof:

Since the set

{(t,tA)Jt£T}

is open in

TxS, if we fix t, then the section. tA is open in S. Continuous divisors thus have the important property of translating open sets into open sets. Corollary 2.8 .

If T is a set of left cancellers,

andcontinuous left divisors,

relative to T, then it is a

set of continuous left cancellers relative to T. Proof:

If the correspondence

s -^4.('k*'ks) 11 € T} is

open in TxS, then it is certainly relatively open in K, so we obtain the corollary by combining both parts of theorem 2.6

. Theorem 2.9 .

If S^ and Sg are two topological groupoids

whose elements are continuous left divisors, then every ele­ ment of the topological groupoid

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11

S^xSgis a continuous left

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2 .1 0

divisor. Proof:

This follows either directly, or from the

previous theorem and the fact that if from A^ to B^, i=-l,2, A 1xA-2

to

then f^xfg

f^

is an open mapping

is an open mapping from

B-jxBgLet every element of the groupoid-

Theorem 2.10

with-topology S he a continuous left divisor and let S ’ be a continuous open homomorphic image of S.

Then every element

of S ’ is also a continuous left divisor. Proof:

The theorem may be proved either directly

or as follows. S onto S 1.

Let f be the continuous open homomorphism of By theorem 2.6, we must show that the mapping

y 1 — ^ x * ,x’y ’)[x* £ S f} of S ’ into S ’x S 1 is open. this is the result of three mappings,

y*

However

f“^(y’),

y-^?x,xy)j x€- s} , where y (f(x),f(xy)) = U ' j X ’y'), where x ’ =f(x); from S 1 to S, from S to-SxS and from SxS to S ’x S 1, each of which is open. We now investigate the embedding process which is fun­ damental to each of the cases "under discussion.

If we have

a set S in which is defined an equivalence relation, then the *v*

set of all points in S, each of which is equivalent to some point of a given set A, will be called the saturation of A and will be denoted by contained in satA

satA.

The set of equivalence classes

will be denoted by [A-].

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In a like manner

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2 .1 1

the equivalence class containing the element s is denoted by [s] and the set of points therein by [[satA]] = [a ] £ £s ]

and

sat s.

[sat a3 = £a]]€£sj .

Thus We shall say

that an equivalence relation is open if satA is open whenever A is open. The natural topology induced on the set of equivalence classes [l: p52] is the finest topology -which is such that the natural mapping

a-^£al

is continuous.

[sj are the sets £a J' where

satA

Thus open sets in

is open in S.

It does

not in general follow, that the topology in [S] is separated. We shall verify that this is so in each particular case. Theorem 2.11 .

Let S be a topological groupoid and

let T be a set of left cancellers in S.

In the subset K

of TxS, consisting of the elements of the form (t,ts),t€T, MWJMJt! >»WWWiai^A W IISW )IMHgmi>a»MW»WM!B'Wiro(t,ts), and (t,ts)->

[t,ts}, from S to 2 and from K to [Kj.

Thus the corres­

pondence is a homeomorphism if and only if the mapping s sat(t,ts) is open and this, by theorem 2.6, and since sat(t,ts) = {(t,ts)| t€T_[ is equivalent to the statement that |

every element

\

to T.

of T is a continuous

left canceller relative

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3

Alternation Tooological G-rouuoids. If in the

Euclidean plane we denote by ab the mid-point between any two points a and b, then it Is clear that this operation has the property of alternation, because of a simple geomet­ rical property of a quadrilateral.

In fact if the plane

is in no way restricted, it. becomes an alternation topolo­ gical quasigroup under this operation.

Any convex set in

the plane will be a topological cancellation groupoid.

If

we take, for example, a circle of unit radius in the plane, then in order that the equation ax = b

should have a solution

when a and b lie inside the circle, we must adjoin the points inside a concentric circle of radius three.

If we repeat

this process a countable number of times, we obtain the whole plane, and In this way we have effectively embedded a topolo­ gical groupoid in a topological quasigroup.

In the general

case one might therefore expect that an alternation cancel­ lation groupoid will generate a quasigroup, by means of a countable number of extensions.

This is in fact the case.

iiany more interesting examples of alternation groupoids are given by Sho lander £7]. We shall be concerned here with the more general case where it is not assumed that every element is a canceller. We begin by recalling some theorems of Sholander If S is an alternation groupoid, let L (R) denote its set of left (right) cancellers and Lw (R*) its set of left (right)

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"cancellers.

The same notation will be used with subscripts

also. Theorem 5.1 .

{7: 4,2 - 4.1C>]

If S is an alter­

nation groupoid and if we set up the relation (a1,ag) in L^xS whenever (a^a^)b^ = (a^b^)a^,then this is an equivalence relation, and the set [L"x S3=S- j_ of equivalence classes be ­ comes an alternation groupoid under the operation £ai,a23[bi,b23 = [a-jb^agbgl . [u,uaf],

S is embedded in

Under the correspondence a — ^ and If"' is embedded in

if a is in If* and b in S, the equation solution

ax = b

while

has a unique

x = [a,b^ in S-^.

S-j_ is called the first left extension of S. vacuous then

S

and

is vacuous.

first left extension of 2^, n ^ 1,

If If* is

If

is the

we obtain an increasing

sequence of alternation groupoids, whose union S«j has more interesting properties mentioned in Theorem 5.2 .

[7: 4.ll

An alternation groupoid

S may be embedded in an alternation groupoid manner that If" is embedded in

in such a

and every element of L^"'

is left proper. is called the left extension of S. may find a right extension of S.

Similarly we

Sholander proves that

the left extension of the right extension is isomorphic with the right extension of the left extension, and he calls either

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3.3 one S' cjooO

*

Tills t reives jJ

Theorem 5.5 .

[7: 5.4j

An alternation groupoid

S may be embedded in an alternation groupoid manner that the ‘"'cancellers of Soooo

in such a

form a quasigroup con-

taining the ‘"cancellers of S. In particular we have

[7: 5.5] that an alternation group­

oid S may be embedded in a quasigroup if and only if every element is a canceller. As might be expected we can find a "workable'* topology in the extensions only by putting further topological restric­ tions upon S, which are carried over to S-^, and which make the equivalence relation in L*"*xS open.

One set of conditions

is the openness of If"* and the continuity of divisors.

This

is the main concern of Theorem 5.4 .

Let S be an alternation topological

groupoid in which every element is a continuous divisor and L* is an open set. (1) (2)

Then:-

the equivalence relation in L‘"xS is open, is an alternation topological groupoid in which

(3)

every element is a continuous divisor,

(4)

1^*" is open in

(5)

the mapping (a,b)->x = [a,b} where a x = b , from L'"xS

and

to. S-^ is unique, continuous and open. (6)

Also

L" is embedded in

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3 .4

(7)

S is embedded in

(8)

S is open in

Proof: lation (a-^ag) L*'*‘xS is open. (bi,bg)e satA. b ^ a , or

and

.

We prove first that the equivalence re­ (b-^jbg) if and only if (a^a^Jbg = (a-jb-j^ag in Let A be an open set in L*xS.

Let b =

Then there is an a -(a^,ag)€ A

(a-^a-^)bg = (a-]_b2 )3 .3 *

such that

Since Lw is open in S, we

may choose HCa-^)* H(ag) in S, where iKa-j^^L* and ECa^xETCagJc A. Consider the expression (a^b-j^agnc.

Since a-jb-^ is a con­

tinuous left divisor, we may choose I'Ua^b^) such that if x 1 is in lUa^b^) then there is an Slow choose I'Ub^) ^ L^

such that

in ^(ag) with x ’ag's; c. a^KCb^)£ ^(a^b^).

consider the expression (a^a^Jbgsc.

The set a^lKa^) is an

open srt of S containing a^a^, by corollary 2.7. is a continuous right divisor, we may find IT(bg)

Since bg in S such

that if bg* is in ITCbg) we may find an a-j_rin lUa^) (a^a^Obg* = c.

Also

Put I'I(b) = Etb^JxiUbg) .

with

Then if b ’^ C b ^ b ^ )

is in U(b) we have that a^b^1 is in a^h'Cb^) and we may find a^* in Nta^) and a2 * in IT(ag) such that ( a ^ b ^ a g 1=. c = (a^a^1)bg* This proves that b ' ~ a ' 6 A L7: 4.33,

bhat satA is open,

so that the equivalence relation in L^'xS is open, and this is (1). We observe that since L* is open then L* is also an alternation topological groupoid element is a continuous divisor.

(7: 3.lQ in which every Thus by theorem 2.9 and

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3 .4

its dual, the same is'true of L*xS. that the natural mapping of L “xS onto

But we have just proved S-^

is a con­

tinuous open homomorphism, so that S-j_, by theorems 2.5 and 2.10 is an alternation topological groupoid with continuous divisors, if we can verify the separation axiom.

This we do as follows.

Let \ei\ and [b] be elements of S-,, where £aj * £b] . Thus if a =.(a-,,a2 ) and b = (b-^bg) then (aiai)t>2 = ^alb l^a2* But vie may find H{(a1a^)bg^ and IT a-jb-^)a2\ in S with a vacuous intersection.

Since S is a topological groupoid, we may

choose H(a.,) and IT(b^) in L"* and l\T(a2),N(bg) in S, such that

{H(a1)LT(a1)} L'(b2)^ U{(a-La1)b2}and ^(a-^UCb-j^lKag) £ ll£(a1b1)a2^. Thus £lT(a, )xlT(az)j , ^2f(b t)xN(bz)j are open sets in S^ contain­ ing £a] and £b] respectively, because the equivalence relation is open, and they have a vacuous intersection, because if we suppose to the contrary, then we may find (a^’jag*)

in

iT(a^)xH(a2) and (b^’jbg*) in IT(b^)xlT(b2) such that a ’^ b 1 or ^a^ tai ,)^2I “ ^al ,^l* ^a2*

"kkis ^-s a c033*1*adiction.

We

have therefore established (2) and (3). Since L-j*' = Ob'bcL^ L"xL

fV: 4.6]

is open in L"xS, so that

and Lv is open in S, then is open in S^_, which is (4).

If a is in I?c and b in S then the unique solution x in Sn of the equation ax But the mapping

b

may be written x = £a,b}

Jj7; 4.103 *

(a,b)-> ja,b] is continuous and open from

L'“xS

to S^_, since the equivalence relation is open, so we

have

(5).

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5 .5

The correspondence with a subset of

a-^£u,uaj is an isomorphism of S

[7: 4.9^ .

This is also a homeomorphism

of S with {u,uS^ because by corollary 2.8, each element of L’“ is a continuous left canceller, use theorem 2.11 & : 4.8] .

relative to IT, so we may

Thus S is embedded in S^.

proves (7) and therefore (5).

This

Since every element of L’"* is

a continuous divisor, it follows from theorem 2.6 (1), that the set

§u,uS)\ uCli^j

is open in L'icS

is open in S^, which is

and hence that S

(8) .

We observe that parts (2),(3) and (4) of this theorem show that the topological restrictions imposed upon S are inherited by the first left extension S^. apply the same theorem to

and so on.

We may therefore 3y applying this

theorem n times we obtain Theorem 5.5 .

If S is an alternation topological

groupoid with continuous divisors and If'* open, then the same is true for each S^, while each subset of

n > 0 , S0 ~S.

If we put Sgf,

is embedded as an open

xOO \J,

as in theorem 2.1.

then we may define a topology in Because each

is open in

then So,, is an alternation topological groupoid, from theorem 2.4 j7: 4.lj .

It may be verified directly that every element

of Soo is a continuous divisor. since L* — \J|In*

Also

-*6l»

-

is open in S^,

each 1^* is open In Sn .

20

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5 .6

Thus ue have Theorem 5.6 .

If S is an alternation topological

groupoid with continuous divisors and L* open, then S may "be embedded in an alternation topological groupoid S^ , ■which is such that every element of

is a continuous divi­

sor, L^' is open and every element of The groupoid

Is left proper.

is called the left extension of S.

In a similar way, If R* is open in S, we may find the right extension of S.

Y!e may also prove that the right extension

of the left extension is isomorphic and homeomorphic with the left extension of the right extension.

For this we

need the following theorem, in which we employ the notation of Sholander £7: 5.1]. Theorem 5.7 .

If S is an alternation topological

groupoid with continuous divisors and both L*' and R*' open, then the first right extension of the first left extension (Si q )o i

is isomorphic and homeomorphic with the first left

extension of the first right extension (Sq x ^i o * Proof:

Sholander has established the algebraic

part of this theorem [7: 5.1].

Since the result is trivial

if either L** or R* is vacuous, we consider the case where both are non-vacuous and hence L5,:‘= R*

[7 : 5.7].

sets up a correspondence between x in (S^q )q ^_ and y

Sho lander in

(Soi)io> by means of the sequence of four correspondences between the spaces (S^o^Ol* RlO*xSio > L*::'xSxL*xS = R*xSxR*xS ,

-

21

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3 .7

|

V | £■■ |

i

Sq-jsRq^,

(SqiJ^o

defined in the following way.

in (S^o^oi* ciloose (**»p) in Rio^xSlO s-uck tb-at xr = p, then jj,

choose (a,c,b,g) in L^xL^xL^xS such that ar = c, b p = g , then

|

(s,q) in ^ o i ' ^ O l is determined by

I

^01^10 ^7

[•

t

Given x

ft

sa = b,ac-=g

and y in

Sholander shows that this is an isomorphism of

1(

v

by using the equality L"'*' and R"“ in each group considered.

I:

A

To see that it is a homeomorphism, we need only observe,

£

fe

Si; f t t? -

from theorem 5.4 partC5) and its dual, that each of the con­ stituent correspondences considered is either a continuous open homomorphism or is the reciprocal of such.

Thus open

sets correspond to open sets in both directions. 7tre now see

[7: 5.2] that if we make m first left exten­

sions and n first right extensions, the result is independent C to whithin homeomorphic isomorphism ) of the order in which these extensions are made.

We thus denote the resulting

topological groupoid by

♦

The next step is to prove

that

is independent of the order of the extensions.

This is

done as follows.

in He©o*X"xSx2b2 ^ ' because any two

equivalence classes in the neighbourhoods on the left hand side may be written

[_a^!,ag>3 ,£b^’,bgQ , where a ^ ’^HCa-^),

a g ’^lUag), bj* ^KCb^), b g ^ H C b g ) , and therefore there exist xp*£ iUx^), x ^ ^ l U x g )

such that

xi*a2* = ^ ’^l* * so " fc3ia'k

[al ‘,a2 ^ B>l**b 2 0 ~ [ xi lal I>x2 ,b2 ,-le LK ^x lal#x2^23 *

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132113

4 .2

is a topological semigroup, if we can establish that it is a separated space. To do this let b = ( b 2 _,b2 ).

[aj ^[b}

in

Thus if we choose

xa^ = yb-^, then

is open, we may find

both il(a^)

and

x' £ lUx) a

b^_

N(b^)

with

y* £iUy)

y

in

C

such that

S

By continuity, and since

C

in

and

C

and

N(ag),mbg) in

If(y)IHbg)Q U(ybg).

Since

are continuous right divisors, we may choose such that if

16 hC a-^)

x ’a^1 = xa^> and if

with

a~b,

N(xag), NCybg) in

EF(x), h(y)

iJ(x)hT(ag) 4 H(xag)

a-j_ and

and

Thus we may choose

with a vacuous intersection.

where

x

a«=(a^,ag),

xag^ybg, for otherwise we would have

which is false. -

S

S^, where

y b ^ y ’b-^1.

then there is an

b-^’^KCb^) then there is

Then

[H(a-j_)xN(ag)l and

are open sets of equivalence classes containing [a}

and

[b"]

respectively, but with vacuous intersection,

for the following reasons.

If we suppose that some equivalence

class is common to both, then we may find iUa^JxNCag),

(b-^jbg*) in

However there exist x ’a ^ ^ y ’b ^ ’. x^g^y^Dg*, fact that and

NCb^xUCbg), where

x* 4 K(x)

and

y ’€H(y)

in

a ,~'bI. such that

If we can prove that this implies that then we have reached a contradiction of the

KCxag)/\Er(ybg) = 0.

x ’a^’= y ’b ^ 1

uag’srvbg*.

Thus we prove that

implies that

means that there exist and

( a ^ S a g 1)

u

x ’ag* ss-y^g1.

and

Choose

v

r,s

-

28

in in

C C

a ’/^b1 But

such that so that

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a ’^ b *

ua-^1 =.vb^*

ru~sx*.

Then C

rvb^’— rua-^1 = sx'a-^1 = sy’b ^ ’, whence since

we have

rv — sy ’.

whence since

s

Also

is in

C

3

x^

2

we have

b^* is in

’=-rua2 *— rvbg 1 = sy’bg* x ’a g T

contradiction being confirmed, we know that [a 3 and Lbl stay be separated by open sets, and

S-^

is therefore a topological

semigroup. The set of cancellers C ^ = [CxC3

and

CxC

is open in

is open in

CxS.

because

Then

C^

logical group because the inverse operation in from the reversal of coordinates in

is a topo­

C-j_ is derived

CxC, which is obviously

a continuous operation. That theorem

2

S

is embedded in

S-^, follows immediately from

.1 1 , since the embedding isomorphism is

and every element of relative to

C

a->[u,ua3

is a continuous left canceller,

C, by corollary 2.8.

'we proceed to generalize this theorem by showing that we may just as well work with ideals as with the whole semi­ group.

hore precisely we say that a subset

semigroup

S

is an ideal of

S

if

S0 S< SQ

S0 and

of a SS0£ SQ .

Using this concept, we may weaken our topological postulates. For this we require the Lemma 4.5 S, such that

If SQn C ^ 0

and if

of the semigroup

SQ , then

Proof:

That

SQ

is an ideal in a semigroup CQ

is the set of cancellers

C0 — S0r\C.

CaS0 ^C 0

is easy.

Let

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c

be in

4 .4

CQ

and let

Choose

u

are in

Sc

a,b in

be elements of

Sq a C , then

so that

S.

Suppose that

cau = cbu.

and

bu

Thus

c

is

Similarly we may show that

c

is

However

au=bu, from which

a left canceller of

S.

a right canceller of

au

c a e cb.

S, and therefore

a = b.

c € C , or

C0 = S 0aC.

This leads to the Theorem 4.4 . SQ an ideal in

Let

S, such that

S-^ generated by

S

S

be a Dubreil semigroup and

Sot\C=£0.Then the semigroup

(according to theorem 4.1)

with the semigroup

SQ^

Proof:

generated by

is isomorphic

SQ .

Vie observe first that

SQ

does generate

a semigroup. SQ-j_ in the sense of theorem 4.1, since also a Dubreil semigroup. find

u,v£ C

such that

lemma 4.3, so are a € S0 , b £ CQ then if

wu

w£C0

then

is

In fact if a , b £ C 0 , then we may ua = vb.

and

we may find

S0

But if

wv, while

wua=wvb.

u4C, v € S

w u g CQ,

w € C Q , by the

so that

w v € SQ , while

Also if ua=vb,

But

wua=-wvb.

It will be necessary to distinguish between the equiva­ lence relations in

CxS

and in

0oxSo , and we shall do this

by denoting the equivalence classes in We set up a transformation from as follows.

Let

w

in

0oxSQ

S-^ = [CxSl

diagCox0o

to

be fixed;

find it convenient to denote the coordinates of same letter, S^,

define

i.e. PCa-3

w = (w,w).

If

> so that

by

£

SD^ — Cc0xS030 we shall w

by the

£a^

is an element of

F[a]

is an element of

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Sq i -

The function

from

onto

then

[wa]G = [wb]0

F

is single valued and one to one,

S0^> because i.e. in

w

is fixed and if

wa~wb

in

follows that

a^b

CxS.

because with

a = (a^,a2 ), b = (b^,b2 )

x^wa2 = XgWb-^-

x -^^Cq

C0xS0 , from which it

It is also an isomorphism

P [a] P[ bj = [wa] 0 [wb] 0 — where

FlaJ=- P[bj

,

in

CxS

,XgWbg ]Q ,

x^

S ^

and

Fit a][ b3} = P [yia^ygbg] = [ w y ^ , wygbg] 0 ,

where

7ia2 ~ 3 r2b l5

such that

Yi€ C,

rxj_w — swy^

y2 £ S.

Choosing

r,s

in

0o

we have

rxgwb ^ =-rx^wag — s w y ^ g ^ swy2b1, whence since

b^C,

rxgTf — swy2 .

[■wy1a1,wy2b2J0=Jx1wa1 ,x2wb2l6 ,

This proves that

and that

P

is an isomorphism.

We may extend this result to topological semigroups in the follov/ing Theorem 4.5 .

Let

semigroup, and let intersects ted

S

C.

S0

S

be a topological Dubreil

be an ideal in

S

such that

Then the semigroup-with-topology

SQ S^

genera-

Is isomorphic and homeomorphic with the semigroup-

-with--topology

So1

Proof:

generated by

SQ .

We use the isomorphism of the previous

theorem, but we observe that the elements in the equivalence class

F(V]= [wa]Q

are precisely

any element which is equivalent to equivalent to

a

in

CxS

i.e.

-

31

(sat a) wa

in

n C0xS0, C0xS0

F£a] =[(sat a)

o

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because

is also Cq X s J

q

.

Any open set

A^_

of S-j may be written

is a saturated open set of which proves that Bq1

FA-^

Bo

prove that

B

CxS.

CQxS0,

w

Conversely, if

C0xS0

and

£b 3 = F“-*-B0-^

CxS.

Let

wb£ B0 .

b

B

so that

b

be an element of

are isomorphic

B.

However the left trans­

is an interior point of F

is a

it remainsto

so

is a continuous transformation from

This proves that Sc^

then

A

we may write B o 1 =.[B01 o ={b ,v(Co x So }3

Thus

is open in

w & d i a g C oxG0

lation by

So1

where

FA-j=F[.A3 — [Aa (Cox So )1 0

Sq I*

is a saturated open set of

saturated set in

Since

Then

is open in

is an open set of

where

CxS.

A ^ = C a 1,

CxS

B, or

is a homeomorphism, so that

to

B

S-j_

is open. and

andhomeomorphic.

Using this we may reduce slightly the topological assump­ tions made on the semigroup in theorem 4.2. Theorem 4.6 . group, and let

S0

Let S be a topological Dubreil semi­ be an ideal in

S, which intersects

Let

SQ

have continuous divisors in

let

CQ

be open in

of CQ

S, relative to CQ .

generates a topological semigroup

Proof:

relative to

SQand let all elements

continuous cancellers in

and such that

SQ

S^_

S

conditions of theorem 4.2,

SQ

S0

S

is embedded

C-j_ is a topological group containing Since the semigroup

SQ , be

Then

in which

C.

G.

satisfies the

generates a topological semi­

group, which by theorem 4.5 is isomorphic and homeomorphic with the semigroup-with-topology

generated by

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S.

Thus

I | * I I

is a topological semigroup and

is a topological

group. We define equivalence classes in CGxS

l

section of the equivalence classes in CxS

I

0oxS.

to be the interwith the space

We denote the equivalence relation thus set up in

I

C©xS

by

[.

thatthe equivalence classes in

(J0xS0

:

of equivalence classes in

with the space

C

Iq*.

p[a] Q =-Ea]0 ’. onto

Prom the proof of theorem 4.5,

CxS

we know

are intersections 0oxSo .

This defines a transformation from

£0oxSjo 5 because

SQ

is an ideal.

Put

£C0xS0^ 0

This transfor­

mation is continuous, by definition of the equivalence classes and is open because the two spaces are homeomorphic. with

(CoxSl0 ‘.

Thus

and

[0oxSo3 Q

is a homeomorphism of

However by theorem 2.11,

phically embedded in is a

P

[CxS^

S

is homeomor-

[C0xS]0 ’, because every element of

continuous canceller in

S

relative to

S is homeomorphically embedded in

[C0xS0]0

CQ .

are connected, then so is

S^_.

and in

Also if

SQ

C0

Therefore

We now remark that if in the above theorem both CQ

[Co^ol 0

S^=[CxS3 SQ

and

has any one

of the properties (a) locally conpact,(b) separated, (c) locally connected, then

has the same property.

• Since any commutative semigroup is immediately a Dubreil semigroup, then the above theorem applies to commutative topological semigroups yielding a generalization of theorem 3.10.

In particular we have

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4.7

Theorem 4.7 .

If a commutative topological cancel­

lation semigroup

S

every element of

S0

to

SQ

which is such that

is a continuous divisor in

SQ, relative

SQ ,

and a continuous canceller in

S, relative to

S

generates a topological group

G

then in

contains an ideal

and

S

SQ,

is embedded

G. That the consideration of the ideal

Sc

in

S

does

indeed give us something more, is illustrated in the following example.

Let

S

be the semigroup of real numbers greater

than or equal to one, under the operation of addition, and with the usual topology. greater than one.

Let

Then

SQ

SQ

be the set of real

is obviously an ideal in

Every element of

SD

is a

continuous divisor in

every element of

SQ

is a

continuous canceller in

tive to

SQ .

numbers S.

SQ,while

Vfe know then, by theorem 4.7, that

S rela­ S

gene­

rates and is embedded in a topological group, which is of course, the group of reals under addition. element

1

in

S

However the

isnot a continuous divisor in

S,

so

without this generalization we could not say anything about the topology of the group generated by the trivial semigroup S.

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5.

Compact Semigroups . In this section we estab­

lish. the fact that any compact topological cancellation semi­ group is already a compact topological group, so that the problem of embedding does not exist. Theorem 5.1 .

For this we need the

Let G be a topological groupoid and

let Q be a subset of G, which is such that Q is an abstract quasigroup, every element of Q is a canceller and Q, is com­ pact [l: p59} .

Then both Q, and Q are topological quasi­

groups . Proof;

Let a and b be any two elements of Q,-

Since Q is a compact space, its topology is uniform [l: pl07j so the neighbourhoods of a point p will be denoted by where A

is independent of p.

We may then

choose directed sets {a^ [ot € & }

and jb*\oc e

ging to a and b respectively.

Since G is a topological

groupoid, then the directed set [a^b^ | €- A } to ah [8], so that ab is in 5-

in Q converges

Also the pair a

defines unique elements c ^ jd^ The directed sets {c^joiG A }

in Q conver­

,bx

in Q, such that a ^ - b ^ c ^ s d ^ b ^ . and fd^ } * £ A]j

are both in

the compact space Q, and therefore have cluster points £93. Let c be a cluster point of {c^iot.^ A.} a^rbc.

Then we may choose Y in A

is vacuous. Thus a

But we may find

and suppose that

so that

(a)n[ys(b)7K (efl

so that c ^

is in V^(c) .

€■ V CC (a)C VxO (a), and bOCcoC € V* (b)V-jwr (c), which is a con-

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5.2

tradiction.

Therefore a=bc, and c is a -unique solution

of this equation because of the cancellation law in

Simi­

larly there is a unique element d in Q such that a = db and this proves that ^ is a quasigroup. To prove that Q is a topological quasigroup, we repeat the construction of the previous paragraph, except that we allow a^ and b^ to lie in ^ and use the fact that a directed set in a compact space, with a unique cluster point, must converge to that cluster point.

Finally since Q is a sub-

quasigroup it is also a topological quasigroup. If we assume the associative law we have Corollary 5.2 -

A compact topological semigroup,

which is an abstract group, is also a topological group. We need also the following theorem which will be stated in a general form so that it can be used also in chapter two. Theorem 5.5 .

If T is a compact topological semi­

group, and if S is a subsemigroup of T, which satisfies the pseudo-commutative conditions (i)

If a,b

are in S

there existx,yinS such that ax = by,

(ii) If a,b

are in S

there existx,yinS with ab = xa = by,

then the set

G - sOtoo S3

is a non-vacuous tooological group. w

Proof:By the hypothesis sets {15)s € A } where vacuous intersection.

A is

(i)the collection of a finite subsetofS, have a non-

Therefore, by the compactness

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[l: p6cf|

5 .3

G is non-vacuouc. If we set up an ordering in S, by saying that a > b if and only if there exists a c in S such that a«=bc, then S becomes a directed set under this ordering, because of (i).

The set

G is then the set of cluster points £93, of the directed set

S in the compact space f.

Because T is a compact space, its

topology is a uniform topology, so we denote a fundamental system of neighbourhoods of a point p by {V^CpHe^ 6 -A\

with

/\ independent of p.

In the product set

SxA

we set up the ordering (a,

(b, p. ) if an only if a > b and

oC> p

, so that S x A

)>

is also a

directed set. Let a and b be in G. given (s, s’ as,oc£lro s’

111611 as„
s and "bSjoC € T^Cb). choice of ©s, ^ in ^ ( a ) .

in S

Row we make a fresh

in S so that a s> b s,^ bg, ^

and agjet is

By postulate (ii) this means that we may write

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5 .4

as,cc =fcS,oc b s,K cs,e,=ds , ^ b s,ec'bs,oc where cSj>c, d S5eC are in S.

Again by (ii) b 3>e(.cS}ei > s

directed set point.

and s*

Suppose that a^bc.

Then by continuity, we may choose p in

/\ so that 7/g(a)A[7p(b)V|a(efl is vacuous.

given t in S, we may choose

a^ ^

^h©

b Sj0

0

,

If we put

in the strong

={x^,... ,xn ; £.} ,

and denote the set of all oc

= £ Tl T € £. , [Tx^ l< £

,1

=

1

, ...,n}

,

by

the

is a fundamental system of neighbourhoods

of the zero operator in the strong topology. It is easy to verify that topological space. because if in

&

is an

0

such that

lT*l ^ M

for all f e T 1 .

is a fundamental durected set in 3 C we may find

T^x

to verify that in

so that

But {T*x)fc6 T}

which is complete, so

T^x-^T^x.

It is a routine matter

is. a linear operator,

llci $ K

T^-^Too

. We know also that

is an abstract algebra satisfying

the postulates (1 ) and (2 ) of theorem is a totally bounded subset of therefore bounded in norm Thus if

p> =[2^,... ,xn ; e \

which we have that putting

2

.2 , because (l) if

(X

£, , then it is bounded and

i.e. \ T \£M put

for all

T

in

- { x-^ ... jx^;

G.1^ ^ ^y3 ? a*1*! (2 )

ot. ={Tx-j_,... j^x ^ £.} ensures that

X . , from

if y3 ={5C^,. ••,2 ^; a} ^ - - ^ 3

•

We shall adopt the following definition due to Eberisin [13J .

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3 .1

A semigroup

in

$

set of operators

is ergodic if there is a directed

{t * I* € T* }■ ,

in

£,

(a)

T^x is in (con S )x for all V

(b)

there is an M > 0

(c)

if T is in

f

with the properties and all x in 3 £ ,

such that |TX( ^ H

for all

V ,

then TimCTT* - T ) - 0 , lim(T5T - T) = 0,

where the limits are in the strong topology. Also an. element x in 3 C

is erfeodic with respect to this

ergodic semigroup if there exists a that

y€(con^?)x

and

Ty— y

y = Tcox

for all

T

in

in

$

3C

such

.

Our main theorem is Theorem 5.1 .

Let

plicative semigroup in

be a totally bounded multi­

S

, which satisfies the pseudo-com­

mutative laws (i) such

if S-j_>Sg are in

that

such that cf

then there existS3 ,SA

in

^

then there exist

in

«f

S-^Sg — SgS^ *

(ii) if s-j.,S2

then

$

S^Sg =

are in

“ SgSA ,

is an ergodic semigroup and every element of 3 € is

ergodic. Proof:

We first show that

semigroup.

Let A and 3 be elements of

directed sets A * € A + V^

{A* lot 6 A } and {B^j* &

and

directed set in V-g

is a topological

$

we may choose

€ B+V* . $

Then

}

a

$

in

We may choose x? ,so that

{A^B^jotG A} is a

which converges to so that

.

A3 because,

£ frV-g ,

and

- 50 i

J Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

V^B £

given .

3 .1

Then for

©y3

we have that - A3 = A ctB ot - AetB + A^B - AB

SAJT^+V^B s vY ;f

.

Is therefore a topological semigroup. The semigroup

is compact because it is a totally

bounded set in a boundedly complete topological linear space. By theorem 5.3 of chapter, one, the set topological group. P in con ^ by

is a

By theorem 2.2, there is an operator

such that

PG- — P =GP

for all

G-

Corollary 5.4 of chapter one, if S is in then

SG

and

G-S

are in

in $

.

Then

and 5

is in

and

SP -S(GP) = (SG)P* P=^ P(GS)= (PG-)S - PS. We may therefore put

P — T-$

trivial directed set, satisfying order that

x

Sy =■ SPx =■Px ~-y

element of

GC

obtain

a

the conditionsrequired in

should be an ergodic semigroup.

$

Finally if and

for all Y and we

is in 3 C for all

then S

in

y=Px $

is in

(conj^x

so we have that every

is ergodic.

We may say a little more about the operator P, as in the following Theorem 5.2 .

The following statements are equivalent.

(1)

P is in

(2)

P = E , the identity of

(3)

. .

consists of one element.

. - 51 -

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3 .2

(4)

The directed set £

converges in

ۥ .

(5)

The directed set $

converges in

£, to S = P .

Proof:

(1) implies (2), because P^ = P

can have only

oneidempotent element

which is the identity.

(2) implies (3), because for all

G

in

^

and a group

G =BG = PG ~ P = E

. (5) implies (4), because the directed set

inthecompact

S'

space

so that it converges to

S

has a unique cluster point

S,

E.

(4) implies (5), because if the directed set is