ane transformations, f, of K into itself. Then there is a common fixed point FIXED-POINT THEOREMS FOR COMPACT CONVEX SETS

FIXED-POINT THEOREMS FOR COMPACT CONVEX SETS

BY

MAHLON

M. DAY

1.

Historical

remarks

L.

E.

J. Brouwer [1]

proved the well-known

Brouwer

fixed-point theorem.

Let K

bean n-cell,that is,ahomeomorphic image of ann-dimensional cube.

Let

_f

be continuous function from

K

into itself. Then there is a point

P

of

K

such that

f(P)

P.

Schauder

[9]

extended the domain of validity of thistheorem by demon-strating the Schauderfixed-point theorem. If

K

is a compact convex subset of 8 normed linear space

X,

and if

f

is a continuous transformation which

carries

K

into itself, then there is at least one point

P

of

K

left fixed byf;

that is,

f(P)

P.

This wasgeneralizednext byTyhonov

[10]

when heshowedthatSchauder’s proofcould be adaptedto prove the existence ofa fixedpoint evenif

X

is a

locally convex lineartopological spaceinsteadof anormed space.

It is clear that if

P

is fixed underf, then it is also a fixed point of every

poweroff,

_f2

f

f,

f3

f

o

f2,

_{andsoon;thatis,}

P

is fixedunder the smallest semigroup ofoperatorson

K

which includes

_f.

Inthesame way,

P

is8fixed

point of everyoneof thefunctions

_f

of family

F

offunctionsfrom

K

into

K,

if and onlyif

P

isalso a fixed point of every finiteproduct,

()i<_n

fi,of

func-tionsfromF.

It

is notknown, even

_for

the one-dimensionalcase, whether every two

com-mutingcontinuousfunctions from

K

into

K

sharea commonfixed point

(see

the research problem of Isbell

[6]).

Thisadds to the interest and value of the generalization ofaspecialcaseof Tyhonov’s theorem dueto Kakutani

[7]

and

A. A.

Markov

[8].

KUTANI-MARKOV

THEOREM.

Let

K be a compact convex set in a locally

convex linear topological space, and let

F

be a commuting family

of

continuous,

ane

transformations,

f,

of

K

into

_itself.

Then there is a common

_fixed

point

of

the

_functions

in

F;

that is, there is an x in

K

such that

f(x)

x

for

every

finF.

As

intheTyhonov theoremweobservethatitcosts nothingin thistheorem toreplace

F

by

2;(F),

the smallest semigroup ofcontinuous,affine

transforma-tionsof

K

intoitself which containsF.

In

this casethe commutativity of the family

F

iscarried to the semigroup

2(F),

sothe theorem aboveisequivalent to that obtained by replacingthe word "family"by "semigroup".

Thepropertyof commutativityisnotshared byall semigroups.

We

discuss here another property which all commutative semigroups have andsomeother

Received November2, 1960.

semigroups

have,

andwe giveanewproofoftheKakutani-MarkovTheorem

whichextendsto this wider class of semigroups.

THEOREM 1.

Let

K

be a compact convex subset

_of

a locally convex linear

topological space

X,

and let be a semigroup

(under

functional

composition)

of

continuousajne

_{transformations}

_of

K

into

_itself.

_If

,

whenregardedas an

abstract semigroup, is amenable, or even

_if

it has a

_{left-invariant}

mean, then thereisin

K

acommon

_fixed

point

of

the family

.

Some generalizations of Theorem 1 are indicated in

4;

in particular, see

Theorem 3.

₄

also contains anexample toshow thatleft-amenable cannot

bereplacedby right-amenable andaconverseofTheorem 3.

2. Semigroups and

invariant

means

An

abstract semigroupis aset of elements with anassociative binary law of

multiplication; a semigroup

of

operators or semigroup

_{of transformations}

is a

semigroup of transformations of some set

K

into itself in which the binary operation used is functional composition--that is,

[f

og](x) f(g(x)) for

eachx in

K.

m(2)

istheBanachspace ofall

bounded,

real-valued

functions

on 2, withthe least upper bound norm.

Let

ebe that elementof

m(2;)

for

whiche(g) l for every g in Z.

Ameanon

is an element

ofm(Z)*

such that

(For

general information and bibliography on means see my paper

[3];

in particular, if isamean

_andf

e

m(2),

thenglbf(2)

<-_

(f) _-< lubf(2;).)

The right

[or left]

regularrepresentation

of

2;over

m()

isthehomomorphism

p

[or

antihomomorphism

_hi

defined from2; into the multiplicative semigroup ofthealgebraofboundedlinearoperatorsfrom

m(2)

into itselfby:

For

each

h in 2, ph

[or

),h] is that linear operator defined

by:

For

each

f

in

m(2)

and each g in 2;

[pf](g) f(gh)

[or

[Xf](g) f(hg)].

A

mean on Z iscalledright

[or left]

invariantiffor each

_f

in

m(2;)

and each

hin2;

(p

f)

/(f)

[or/(X f)

/z(f)].

is invariantifitis bothright and left invariant. 2;iscalled amenableifthere

existsaninvariantmean onZ. Ifweexpressthis intermsof adjointoperators

of the linearoperators phor

,,

a mean is a right, or

left,

ortwo-sided, in-variant mean if and only if is a fixed point of every po*, or every

ho*,

or

both,

respectively.

Note

that the amenable groups are the groups called "messbar" in von

Neumann’s

paper

[11]

onthe theory of measure.

Some

recentScandinavian writers

(FOlner

[5])

call them

"groups

withfull Banachmeanvalue".

FIXED-POINT THEOREMS FOR COMPACT CONVEX SETS 587

abelian groupsbyanapplication ofthe Hahn-Banach

theorem;

yon

Neumann’s

paperreferredto above has itforabelianand solvablegroups; my 1942 paper

[2]

onergodicity of abelian semigroups hasaproof for thegeneralcasebymeans

of theHahn-Banach

theorem,

essentially usingBanach’s proof.

THEOREM

2.

Every

A

belian semigroupisamenable.

TheprooffromtheKakutani-Markovtheoremisvery brief.

3.

Proof

of the general

fixed-point theorem,

Theorem 1

Let

y be any element of

K,

and define a linear mapping

T

from

X*

into

m(2;)

by attaching toeach in

X*

the function

_T

on 2; such that

[T](g)

(g(y)) for eachgin 2:.

_{Because o}

iscontinuous, it isboundedon the

com-pact set

K,

so

_T

is in

m(2;)

for each in X*.

It

is clear that

T

is linear from X*into

m(2;)

hence the function

T

dual to

T

canbe constructed to carry each element of

m(2;)*

to an element of

X

*

[T](o)

(To)

for each in

X*.

Let

K’

be the image of

K

under the canonical mapping

Q

of

X

into

X

*

[Qx]()

(x)

forall

_o

inX*.

Let M

be theset of meanson 2:, that is, the positive face of the unit ball in

m(2;)*.

LEMMA

1.

_If

e

M,

then

T

e

K’.

Proof.

Ifoe

X*,

then

[T](o)

(To)

=<

sup T(g) g e

_2;}

sup{(gy)

"ge2;}

-<_

sup{o(x)

"xeg} sup{Qx(o)

"zeK}

sup

{(o)

e

g’}.

This says that

T

is in every half space of

X

*

whichis determined bya

in

X*

andwhichisclosed.

By

M:azur’s

theorem applied to

X

_*

initsweak* topology,

K’

isitselfthatintersectionof halfspaces;hence

T

e

Kp.

From

this lemma we knowthat if is in

M,

then

T

is in

QK,

so

Q-IT

is in

K.

Let

j bethemapping

Q-T

of

M

into

K.

LEMMA

2.

_/f

isin

M

and hisin 2;,thenj(*() h(j()

).

Proof.

Ifg e

Z,

let bethe element of

M

definedby

(f)

f(g) foreach

f

inm

(2;).

Thenforeach

h

in2;

[)*0](f)

O()f)

[)/](g)

f(hg)

[hg](f);

thatis,

*-

hg. Also for each in

X*

Hence

j gy if gis in 2.

Because

j is affine, for each finitemean

_’g

a(g)--that isto say, all

a(g) arenonnegative, all but finitely many are zero, and

g

a(g) 1--it

follows that

jS

j( a(g)()

a(g)j()

a(g)g(y).

But

j(X*Sz)

J(E

a(g)X*)

J(E

a(g)-)

E,

a(g)j(-)

ga(g)h(g(y))

a(g)h(j())

h

(

a(g)j

()

)

(because

hisaffine)

h(jS).

But

eachof the functions

_Xh*,

h,and j iscontinuousin an appropriate sense;

alsothefinite means arew*-densein

M. Hence

for each in

M

j(Xh*()) j(X*(w*-lim

eL))

j(w*-lim

(X*&))

limj(X*(&)) limh(j(S))

h(j(w*-lim

&))

h(j(,)).

From

the lemmawe seeatoncethat if is aleft-invariant mean on2, then

j is a commonfixed point of all hin2:. This isthe desired conclusion forour

theorem.

4. Generalizations:

An

example

(a)

The proofs of the preceding section can be carried through in some cases when thereisnot aleft-invariantmean onthewhole of

re(Z),

because there aremany situationsinwhich asmall subspace of

m(2:)

has appropriate properties and contains all ofthefunctions

_T

eX*. Calla

closed,

linear

subspace

Y

of

m(2:)

left-invariant

iffor each hin 2:and each yin

Y, X

y e

Y;

call aleft-invariant

Y

_{left-amenable}

ifee

Y

andthereis a left-invariantmean on Y: that is, there is an element of Y* such that for each y in

Y,

glb y(2)

=<

t(Y) _-<

luby(2) and

t(Xh

y) tt(y) for each hin 2:.

With these assumptions on

Y

the proofs of

Lemmas

1 and 2 go through unchanged.

(The

fact that finite means are w*-denseinthemeansin

Y*

de-pendson the fact that eachmean in Y* hasan extensionwhich is a mean in

m(Z)*;

this is a consequence of the Krein monotone extensiontheorem; see [4, Chap.

I,

Sec. 6].)

(b) Suppose,

for example, that 2:, whichisset of functions from

K

into

K,

isgiven the topology of pointwise convergence. Then composition, the

FIXED-POINT THEOREMS FOR COMPACT CONVEX SETS 589

each

_T

is continuouson 2;. Clearly

C(2;),

the space of

bounded,

continuous

real-valuedfunctions on 2;, is left-invariant. If

C(2;)

is also left-amenable,

then there is in

K

acommonfixed point of the elements of 2;.

(c)

With this in mind let

A (K)

be the semigroup

(under

composition) of allaffine, continuousmappings of

K

into

K,

and topologize

A (K)

withthe topology of pointwise convergence.

Let S

be any semigroupwith atopology

inwhich multiplicationiscontinuousineach variable,and letrbeacontinuous homomorphism of Sinto

A(K).

Applying the remarks above to 2

r(S)

weobtain the following generalization of Theorem1.

THEOREM3.

_If

thereisa

_{left-invariant}

mean on

C(S),

then

_for

each compact

convex set

K

ineach locallyconvexspace

X

and

_for

eachcontinuoushomomorphism r

_of

Sinto

A (K),

there is in

K

acommon

_fixed

pointp

_of

all the

_{transformations}

in the set r

(S).

It

should be recalled that

Haar

measure defines a left-invariant mean on

any compact group, so thistheorem includes the caseswhere

K

is adiscrete

abelian semigroup or a compactgroup.

The condition of the theoremis sufficient aswellasnecessary. Theadjoint representation

hh*

of the left regular representation over

C(S)

has a fixed point if and only if

C(S)

is left-amenable. For

S

a discrete group this was first observedbyGranirer.

(d)

An

example shows that left-amenable can not be replaced here by right-amenable. This is a consequence of the order in which composition is

carried out.

Let

Kbe the unit interval0

=<

_-<

1, and let

h(s)

for 0

_-<

s

_-<

1, 0or 1. Setting

{ht

0 or

1},

we get a semigroup

inwhichthe product of eachorderedpair of elementsisthefirstelementof the pair.

Every

ph is the identity in

m(2;);

no mean is left-invariant on

m(2;).

Ofcourse, thereisnocommonfixedpointof thetransformationsofthe set because the ranges of the two transformationsare disjoint.

REFERENCES

1. L. E. J. BROUWER, ber Abbildung yon Mannigfaltigkeiten, Math. Ann., vol. 71

(1911), pp. 97-115.

2. M. M. DAY, Ergodic theorems for Abelian semigroups, Trans. Amer. Math. Soc.,

vol. 51 (1942), pp. 399-412.

3. ----,Amenable semigroups, IllinoisJ. Math.,vol. 1 (1957),pp. 509-544.

4. ----,Normed linear spaces, Berlin-GSttingen-Heidelberg, Springer, 1958. 5. E. IOLNER, On groups with_full Banach mean value, Math. Scand., vol. 3 (1955),

pp. 243-254.

6. J.R.ISBELL, Commuting mappingsoftrees,Research Problem7,Bull. Amer.Math.

Soc., vol. 63 (1957),p. 419.

7. S. KAKUTANI, Two fixed-point theorems concerning bicompact convex sets, Proc.

Imp.Acad.Jap.,vol. 14 (1938), pp. 242-245.

8. A. A.M.lKOV,Quelques thormessurles ensemblesabliens,C.R. (Doklady) Acad.

9. J. SCHAUDER, DerFixpunktsatz in Funktionalrdumen, StudiaMath., vol. 2 (1930),

pp. 171-180.

10. A. TYCHONOFF,Ein Fixpunktsatz,Math.Ann.,vol.111 (1935),pp.767-776.

11. J. VONNEUMANN, Zurallgemeinen TheoriedesMasses,Fund.Math.,vol. 13 (1929),

pp. 73-116.