The Analogue of the Group Algebra for Topological Semigroups (Chapman & Hall CRC Research Notes in Mathematics Series)

>

I

2 hl.l ci 8•l.l*l.l-x*l.l*ll> +

1

by Theorem 2.3.

<

--=---.....2 llx8*l.l*l.l-i*l.l*llll

+

o,

lh 8

I

+

lh <x*p> h : (p) h~p) 8 lh <x*p> h :8<'1.1> - h<~>

I

I

This contradicts (*), and hence Tis continuous.

41

It follows that A• is a homeomorphism. In particular S is therefore a locally compact space, by the local compactness of the Gel'fand topology. element 1 (say), then for any

x.~ ~

If in additionS has an identity

S we have

x-~(1)

:= x(l)~(l) = 1 and

A

one can easily note that

X·~ ~

S , and hence that S

is a topological semi-

group.

If S has an identity etement 1 then the fottowing items are equivatent: (i) S (endOwed with the compact open topotogy) is compact; 5. 7 COROLLARY. A

(ii) S is discrete. Proof.

Item (i) implies that M (S) a

compactness of Ma(S)

is compact, by Theorem 5.6.

Now the

in turn implies that I~ Ma (S), (see e.g. Hewitt and

Ross [59, Appendices C.24 and C.25]).

Now from Exercises J.lO(c)(ii), we

have that S is discrete. That item (ii) implies (i) follows trivially from the definitions. We now turn to the question of semi-simplicity of the algebra Ma(S). 5.8.

We recall some standard results for a commutative Banach algebra A.

Let A := {h

~

A* : h(ab) = h(a)h(b) for all a,b

~

A and h is non-zero}.

Every ideal I of A is maximal if and only if I is the kernel of some element of A. The radicat of A, denoted by rad(A), is the intersection of all maximal ideals of A. (We assume this definition also for cases where A is not commutative.)

A is said to be semi-simpte

if rad(A) = {0}.

Thus,

equivalently, A is semi-simple if and only if for every a ~ A with h(a) = 0, for all h ~ A , we must have a equal to the zero element of A. t~e shall use the latter observation in the proof of our next Theorem.

Let S have an identity etement 1.

5. 9 THEOREM.

are equivatent. (i) Ma(S) is semisimpte, (ii) M(S) (iii) S

42

is semisimpte, separates the points of S.

Then the fottowing items

Proof. Suppose item (i) holds. We prove that item (ii) holds. ~t p,v E M(S) be distinct. Recalling that M(S) can be identified with c 6 (S)* we can find f E C(S) such that p(f) ~ v(f). Thus we have pof, vof E C(S) with pof(l) = p(f) I v(f) = vof(l). So Proposition 1.3.3 implies that we can find p E Ma(S) such that p*p(f) - v*p(f)

=I

(pof-vof)(x)dp(x) I 0.

Thus p*p I v*p and we can find hE Ma(S) with h(p*p) I h(v*p). In particular h(p) I 0, so we can extend h to h' E M(S) by setting h'(n) := h(n*p) h(p)

(n E M(S)),

Clearly h'(p) ~ h'(v) and so item (ii) holds. Next we assume item (ii) and prove item (iii). Let x,y be distinct points of s. We can find an open neighbourhood V of 1 with Vx n Vy = + and a non-zero measure v E Ma (S) with supp(v) c V. Hence v*x ~ v*y and we can find hE M(S)A with h(v*x) I h(v*y). In particular h(v) I 0 and . . A h(v*s> def1n1ng X E s by x(s) := h(v) we get x(x) I x(y) and item (iii) follows. Finally we assume item (iii) and prove item (i). Let p be any non-zero measure in Ma(S) and H(S) be the linear span of S in C(S). Then H(S) separates points of S and contains the constant semicharacter 1. By Appendix A.4, we have that H(SA) is (norm) dense in L1 (S,Ipl). Since f fdp defines a non-zero functional on L1 (S,Ipl), there exists g E H(SA) such that

+I

I

gdp ~ o. Hence there exists X E ~ with xdp o. Recalling Theorem 5.3 we have h(p) I 0 for some hE Ma(S) • Hence (iii) implies (i).

f

+

6. NOTES ON REFERENCES

For locally compact topological semigroups the object Ma(S) was first introduced by A.C. and J.W. Baker- ([5] and [6]) and subsequently studied extensively by G.L.G. Sleijpen ([92], [93] and [94]). In particular, for such semigroups S, the results of sections 1 and 2 can be found in [5], [6] and [92]. The extension to include topological semigroups that are not necessarily locally compact is due to Dzinotyiweyi [30].

43

The object Me(S) was first introduced by Dzinotyiweyi subsequently studied by Dzinotyiweyi and Sleijpen

3 are [32] and

[32]

and

[39]

[39].

Section

taken from

from

are largely inspired by the results of

[32)

and was

The results of

while those of Section 4 are taken

[92]

and

[93].

For locally compact topological semigroups S the results of Section 5 can be found in

[s],

[6] and [92].

The extension to all C-distinguished

topological semigroups is taken from [30). That the spaces Ma1 (S), Me1 (s), Ma (S) and Me (S) can be identified with 1

L (S) in the case of a locally compact group has been known for sometime (see e.g. Hewitt and Ross [59) and Glicksberg [45]).

44

3 Foundation semigroups and their generalization 1. INTRODUCTION 1.1 DEFINITION. Let S be a topological semigroup with identity element 1 and such that for each neighbourhood U of 1 (a) x

E

int(U- 1 (Ux) n (xU)U-l)

for all x E S, and

(b) 1

E

int(U- 1v n wu- 1 )

for all v,w

E

U.

If S is locally compact then S is called a stip. The notion of a stip was introduced by Sleijpen (see e.g. [92] and [94]); its importance stems from the fact that every locally compact foundation semigroup with identity element is a stip (see Theorem 1.2), and a very large class of stips are foundation semigroups (see e.g. Section 2). Further, it still remains an open question whether every stip is a foundation semigroup. (Of course every locally compact topological group G is a stip and one can extract various subgroups of G that are stips.) This chapter is devoted mainly to a study of the structure and properties of stips.

In contrast to Chapter 2, most techniques employed in this

Chapter have a large topological (semigroup theoretical) flavour and measure theoretic notions are totally absent in various parts and suppressed where they appear.

The results of this section and sections 2 and 3 are taken

from work of Sleijpen ( [92], [93], [94], and [98]) and those of section 4 are taken from Dzinotyiweyi

([33] and [34]).

1.2 THEOREM. Let S be any topological semigroup with identity element 1 and supposeS is the foundation of Ma(S). Then s has properties (a) and (b) of item 1.1. In particular if S is also locally compact then S is a stip. Proof.

Let U be a given neighbourhood of 1.

By Theorem 2.1.7 we can find

a positive measure ~ E Ma(S) with D := supp(~) compact and D S U. Now item (a) follows trivially from Lemma 2.1.9 (and its right handed analogue). 45

To prove item (b) we first choose a positive measure v C :• supp(v) compact and CC

c u.

€

M (S) with a

Then mimicing the argument used in the

first paragraph of the proof of Theorem 2.1.12, we can find a € supp(v*v) c CC such that v(y c- 1) > 0. So we can find a compact

y

0

-

0

D c y0 C-l n C such that v(D) > 0. Lemma 2.1.9.

Hence DD-l is a neighbourhood of 1 by

But

DD-1 C (yoC-l)C-1 = yo(CC)-1 ~ yoU-1 -1

Hence y0 U

is a neighbourhood of 1.

It is now trivial to note that item

(b) holds. 1.3 Blanket Assumption.

· with identity element

1

FoP the PemaindeP of this chapteP S denotes a stip e~cept whePB the contPaPY is explicitly stated.

2. THE STRUCTURE OF STIPS We start by introducing and describing a subset

s1 of S that plays a pivotal

role in the topological theory of stips. 2.1 NOTATION. We denote by s 1 the set of all x € S such that 1 1 € int(X- x n xX- 1) whenever X is a neighbourhood of x. (Here the subscript 1 in notation 2 • 2 THEOREM.

s1

refers to the identity 1 of S.)

s1 is a dense ideal of

S.

First we show that xy € s 1 given that x € s 1 o.nd y € S. Let lol be a neighbourhood of xy and X a neighbourhood x with Xy ~ W. By definition of s1 we then get

Proof.

1

€

int(xX

-1

)

~

int((xy) (Xy)

-1

S int((xy)tol-1 ) •

{1)

Choose neighbourhoods V of x and U of 1 such that VU C X. -1

are neighbourhoods of 1, item l.l(a) implies that U ((V

-1

Since

v-1x

and U

x)y) is a neigh-

bourhood of y, and so 1

€

-1 -1 -1 -1 int(y (U ((V x)y))) S int((VUy) (xy)}

From (1) and (2) we conclude that xy

ss 1 46

~

s1 •

€

~

s1 and so s 1s

int(W

~

-1

(xy)).

s1 • Similarly

(2)

Second, we show that s 1 is dense in s. 1

£

It is sufficient to show that

s1 .

Let U and V be compact neighbourhoods of 1 with v 2 cU. can find v,w

V such that 1

£

int(V

£

-1

v n wV

-1

By item l.l(b) we

-

),

Then u := wv is in U and

S int(U-1 u n uU-1 ).

(3)

To show that u £ s 1 , let X be a neighbourhood of u and suppose, on the contrary, 1. int(X- 1u). Then we can find a net(x) such that a

X

a

and eventually

-+- 1

(4)

By item (3) we choose a net (u ) c U such that a

-

Since U is compact and xa -+- 1, we may assume that ua -+- u, Thus eventually u = ua x a £ Xxa or x a £ x- 1u (since X is a neighbourhood of u), which contradicts item (4). By this conflict we must have 1 £ int(X- 1u) and similarly 1 £ int(uX- 1). Hence u £ s 1 n U and 1 £ 1 •

s

We will need the following Proposition in Chapter 5. 2.3 PROPOSITION. If J is a dense (right) ideal of S then the interior of (S\J)(S\J)-l is empty. In particular int((S\S 1)(S\S 1 )-l) = ~ Proof.

Suppose we had x

ideal, x(S\J)

£

int((S\J)(S\J)

-1

) n J,

S J and so x(S\J) n (S\J) =~or x

£

Then, since J is a right (S\J)(S\J)- 1 • Conflict:

The second part of our Proposition now follows from Theorem 2.2. The next Theorem says that within the category of dense left or right ideals of S we have that s 1 is the "smallest" such ideal.

2.4 THEOREM. sl

We have {x

£

s = n{J s = n{J

int(S -1 x u xS -1 ) }

s

1

s

SJ c J and

J

S}

s

JS c J and

J

S}

£

47

= n{J

~

S}.

Let x x

£

S : SJS

Let A := {x

Proof.

J

S

£

£

SJ

S : 1

£

£

J

= S}.

int(S-1x)} and B := n{J

SS

A and J c S with J = S and SJ c J.

SJ C J.

1

and

Hence x

£

Then J n (S- 1x) ~ ~ and so

B and A C B. £

B and set I := S\S-1x.

Clearly SI S I and

So I~ S (by definition of B) and in particular 1 ~

I.

SJ c J

We now show that A = B.

On the other hand let x x

and

£

-

int(S\I)

S S~ x

and

X £

I.

Hence

A.

Thus B c A and we conclude that A = B. By definition of s 1 and Theorem 2.2 we have that

It is now trivial to note that the theorem follows. As a consequence we have the following factorization of s 1 • 2.5 COROLLARY. Proof.

Clearly s 1s 1 S s 1 •

and hence sl

s

Now s 1s 1 =Sand S(s 1s 1 )S S s 1s 1 by Theorem 2.2,

slsl by Theorem 2.4.

In the situation of a topological group G open sets are well behaved; for instance for every open U c G and x E G we have that xU, x- 1u, xu-l and -1

U xU are open subsets. For stips this can be totally Jisobeyed even when attention is restricted to compact commutative stips. For an example consider S := [0,1] with the interval topology and multiplication given by xy := min{x+y,l}.

One can easily note that S is in fact a compact foundation semigroup with identity element. Now s- 10 - {0} and Sl • {1}. With these

remarks in·mind the following result of Sleijpen [77] is of great interest. Throughout the proof of the Theorem we ask the reader to keep in mind the fact that sl is a dense ideal of s (see Theorem 2.2).

48

2.6 THEOREM.

Let O,U be open subsets of S and

int((x(U n s1))u-1)

foro aZ.Z.

u

£

int((xu)(U n s1)-l) -1 (iii) x(O n s1) is openJI

foro aZ.Z.

u

£

(i)

X£

(ii)

X £

(iv) (xO)U-l

X £

S.

Then

u n s1 • u n s1 •

is open.

Proof. First we prove items (i) and (ii). Let u £ U n s 1 and choose u1 ,u 2 £ s1 such that u • u1u 2 , by Corollary 2.5. Let u1 ,u2 ,v be open subsets of S such that u1 £ u1 • u 2 £ u 2 • 1 £ V and vu1u 2 S U. We then have 1

int(u 1 u~ 1 )

£

S

int(u(U1u 2)-l)

S

int(u(u1u 2 n s1)-1),

and so by item l.l(a) we have that

and X £

This completes our proof for (i) and (ii). Second we prove item (iii). Suppose t £ x(O n s 1 )-l and choose w £ 0 n s 1 with tw .. x. Let V be a neighbourhood of 1 and W of w with VW co.· By (ii) and the fact that s 1 is dense we can find

v

w(W n s1)

£

-1

n v n s1 •

W n s 1 with vy • w. -1 -1 -1 We note that (tv)V S x(O n s1) for if p E (tv)V then So we can find y

£

x = tw • tvy and so p t

£ £

£

pVy £ pV(W n s 1) £ p(O n S1)

x(O n s 1) -1 •

Hence, with item (ii) in mind, we have that

int((tv)V-1) S int(x(O n s 1>-1>

and so (iii) follows. 49

Finally we prove item (iv). with da

Let d £ (xO)U-l and choose a£ U and b £ 0

= xb.

Since s 1 is dense and 0 and U are open, we can find c £ s 1 such that ac £ U and be£ 0. Now d(ac) x(bc) and ac £ s 1 • Hence

d £ (xO)(U n s 1 )

-1

and (xO)U

-1

£ (xO)(U n S1) -1 or (xO)U-1 = (xO)(U n s 1) -1 •

Recalling item (iii) we are done. 2.7 EXAMPLES.

We mention some examples of stips, (in fact of foundation

semigroups with identity element,) from which the reader can easily note that in general we may have the following situations: s 1 f. S even when S is cancellative;

1

+s1

and so

S is "far" from being cancellative;

{1} n ES is not necessarily a G6-subset of ES, (recall that E8 is the set of all idempotents inS with the restriction topology of S); we have compac t se t s C, D C _ S and x £ S sue h that C-lD and x- 1D are no t compac t ;

s1

is not necessarily open or even a G6-subset of S. (i) S := [o,~) with the additive operation and the usual topology. (ii) Let T be any discrete space with zero multiplication xy

some fixed z £ T. (iii) Let Xi S :=

:~

z for

Take S to be T with an adjoined isolated identity 1. {0,1} with maximum operation and discrete topology. Take

rr x. the product semigroup with i£A

:~

A countable.

1

(iv) Take S to be as in (iii) but with A uncountable. For many more examples of stips we refer the interested reader to Sleijpen [92]. The following exercises can easily be solved now by routine arguments. 2.8 EXERCISES (i) For every e £ ES we have that eSe is a stip with identity element e and (eSe)e = es 1e = eSe n s 1 •

[Note that (eSe)e is to eSe what s 1 is to

s.] (ii) If S and T are stips then S x T is a stip. (iii) If S is a stip and T is a locally compact quotient semigroup of S with identity element, then T is a stip.

50

(iv) Let G1 := {x E S : there exists y E S with xy an algebraic subgroup of Sand G1 = s- 11 n ls-1 • If 1

E

s 1 , then G1 is an open subgroup of s.

If 1

E

s 1 and Sis connected, then Sis a group.

yx

1}.

Then G1 is

(See [92] and [94].)

3. THE ROLE OF IDEMPOTENTS IN A STIP This section is essentially a continuation of the preceding section except that here we focus our studies on the contributions of E5 to the structure of S. We have already seen that s 1 is a very powerful organ of S (see e.g. Theorem 2.6). If in addition 1 has a countable neighbourhood basis inS a lot more can be said.

But then there is a large class of stips without the

latter property (see e.g. Example 2.7(iv)).

Fortunately most of the studies

one can do on a stip with its identity having a countable neighbourhood basis can be extended to a general stip S by use of certain elements of Es the so called IS-isolated idempotents. I t is our intention in this section to lay the foundation for such an extended theory. 3.1 DEFINITION.

An element e in ES is said to be 15-isoZated if {e} is a

G15 -set in eSe n E5 , where eSe n E5 is .endowed with the restriction topology.

We denote the set of all 15-isolated idempotents in S by E~. NOTE.

The identity element of a locally compact topological group and the

identity element 1 of a stip which has a countable neighbourhood base of 1, are 15-isolated idempotents.

The identity element of the stip mentioned in

Example 2.7(iv) is not 15-isolated. We now characterize the 15-isolated idempotents partly in terms of s 1 (see Theorem 3.6). Towards this end we first prove some lemmas whose full generality will mainly be needed in later parts of this book. 3.2 LEMMA. Let K be a compact subset of S and x n{s- 1y n yS-l : y E Kx} is a neighbourhood of 1. Proof.

By Corollary 2.5 we can write x

E

s1 •

Then

= x 1x 2 for some x 1 ,x 2 -1 -1

Theorem 2.6 ((iii) and (iv)) we have that (yx 2S)(xs 1 ) neighbourhood of y, for all y E K.

E

s1•

From

is an open

51

But

and so the compactness of K implies the existence of a finite set {y1 , •• ,yn} in K such that n -1 u (yix2S)x • i=l

Kc

A routine exercise on definitions now leads to 0

S n{(S- 1y

n yS

-1

: y

Kx} where 0 :=

€

n

-1

-1

n ((y.x 2 >s 1 ) n s 1 x. 1 i=l

Since yix 2 € s 1 fori • l, ••• ,n; Theorem 2.6(iii) (together with its right handed analogue) says that 0 is a neighbourhood of 1 and our lemma follows.

Let B be a G6-subset of S containing 1. Then theroe e:r:ist a 6-isolated idempotent e in B and a compact subgroup G with identity element e such that G S B and G is a G6-set in eSe. 3. 3 LEMMA.

Proof.

There is a sequence {Vn} of open subsets of S such that B

.

n Vn. n""l Inductively we can choose a sequence {Un} of open relatively compact neighbourhoods of 1 such that (n

£

z

lN) •

Then one can easily see that H :=

.

n Un is a compact suLsemigroup of S n•l

contained in B. Now taking e to be an idempotent in the minimal ideal of H we have that G :• eHe is a compact subgroup of S with identity element e (see e.g. Theorem 1.2.3). Since G := eHe = H n eSe, His a G6-set in eSe. {e} = H n(Es n eSe) = eHe n Es we have e £ E~.

Since

3.4 LEMMA. Let V be an open neighbour>hood of 1, M the closuroe of a a-compact subset of s, and A a countable sul;set of s 1 • Then theroe e:r:ists a 6-iso Zated idempotent e in V such that MA S eSe.

52

Proof.

Let {Kn} be a sequence of compact sets such that M is the closure

co

Since AS s 1 , Lemma 3.2 implies that we can find a G6 -subset

of u Kn. n=l B such that

co

1 ~ B C V n (n{S-ly n yS-l : y ~ ( u Kn)A}). n=l By Lemma 3.3 we can find e ~ B n E~.

(*)

It is now easy to note that (*)

implies that co

( u

K )A

n=l

n

C

-

eSe,

which by the closedness of eSe implies that MAS eSe.

3.5 DEFINITION. A subset F of S is said to be countably closed if F contains the cluster points of countable subsets of F. The countable closUPe ofF, denoted by x0 -clo(F), is thus given by x0 -clo(F) := {x ~ S : x ~A where A is a countable subset ofF}. Now we observe that for any subsets B and C of S, we have that x0 -clo(B)x0 -clo(C) C x0 -clo(BC). In particular it follows that the countable closure of an ideal is an ideal. (For instance x0 -clo(S 1 ) is an ideal of S.) 3.6 THEOREM. aFe

Let e be an idempotent element in s.

Then the foUO!Jing items

equivalent: (i) e is 6-isolated;

(ii) e ~ x0 -clo(S 1 ); (iii) thepe exists a compact subgPOup G of S such that e is the identity ele~nt

Proof.

of G and G is a G6-set in eSe. That item (iii) implies item (i) follows easily from our definition

of a 6-isolated idempotent. Now applying Lemma 3.3 to the stip eSe we have that item (i) implies item (iii). Thus (i) and (iii) are equivalent.

53

Next we assume item (iii) and prove (ii).

~t

{Vn} be a sequence of open

n V and V+l c V for all n•l n n - n Since (eSe)e • eSe n s 1 , (recall Exercise 2.8(i)), there exists a n € lN. sequence {hn} in s 1 such that

relatively compact sets in eSe such that G =

Let h € n clo{hn : n ~ m}. Then h € n Vm = G and mE:tl' m€:11' e € clo{hk : k € lN} since G is a compact group. This leads to for all n € '6.

e € clo{hk : k € :tl'} c clo{hk : n,k € lN}- -c x -clo(S 1). -

n

-

o

Thus item (ii) holds. Finally we assume (ii) and prove (i). Clearly item (ii) implies that e € xo-clo(es 1 e) and hence e € xo -clo((eSe) e ), by Exercise 2.8(i). So we can find a countable set A c (eSe) with e € A. Applying Lemma 3.4 to e the stip eSe with M :•{e} we find a 6-isolated idempotent f in eSe such that A c fSf. Since fSf is closed we have that e € fSf. Consequently e = f and item (i) follows. 3.7 EXERCISES (i)

x0 -c-lo(S 1 )

•u{eSe: e € E~}.

(ii) If e € E~ then (eS u Se) n ES £ E~. (iii) If 1 has a countable neighbourhood base then all idempotents in S are 6-isolated. (iv) LetS be the stip mentioned in Example 2.7(iv). sl

= {f € {O,l}A

x0 -clo(S 1 )= {f € {0,1} A and hence

(c.f. [92] .)

54

f(x) = 0 for finitely many

X

Then € A},

f(x) .. 0 for countably many x € A}

We conclude this section by giving a proposition whose full generality is mainly needed in Theorem 5.2.8 and is given here to avoid repetition. For our purpose in this chapter we are interested in the first item of the proposition which says that for many stips compact neighbourhoods can be realized as closures of products of compact subgroups and countable subsets. This observation has significant applications in our studies - see e.g. proof of Theorem 4.6.

Let S have a 6-isol.ated identity element 1, F be a G6-subset with 1 £ F and A be a Felatively compact subset of s. Then theFe e:z:ists a G6 compact subgroup G with 1 £ G, G c F and a countable subset B of A such that 3.8 PROPOSITION.

A = GB

(i)

;

..

A· (iii) if 'II' is the nozrma.Zised HaaP measupe on G we have that {f £ C(GA) : x*'ll'(f) = f(x) foP all X in GA} is a (noPm) sepaPable subset of (ii) Gx = xG foP all x

£

C(GA). Proof. Our hypothesis implies the existence of a decreasing sequence {Vn} of open relatively compact subsets of S such that

..

V

C F

n-

and

.

E5 n n vn = {1}. n=l

Recalling property (a) in our definition of a stip, an easy induction argument yields an increasing sequence {B } of finite subsets of A and a n

decreasing sequence {W } of open relatively compact neighbourhoods of 1 such n that, for each n £ 1N, -2 wn+l A

c wn n vn c (B W )W-l n W-l(W B )

-

nn

n

n

nn

for all b

We set G :=

..

n n=-1

wn

..

and B := u n=l

B • n

£

B • n

Then G is a compact subsemigroup of

55

S con~ained in F and has exactly one idempotent, 1.2.3, we have that G is a compact subgroup. To prove item (i) let x € A be fixed and W be Then GW is an open neighbourhood of Gx. An easy using compactness of A), we can find n € :N such

An w-n 1 (wn x) €

Ac

w~ 1 (wnx) and so B n GW ~

Gi.

Recalling Theorem

an open neighbourhood of x. calculation shows that, (by that

c- GW.

Since A c w- 1 (w B), we can find b nn - n b

namely 1.

+•

€

B with n

X €

w- 1 cw b). n n

Consequently GB n W ~

Hence

+ and

we have

Since A is relatively compact and GA is compact, we then get

GA .. GACGGBCGBCGA. Thus A .. GB • Hence A • cGi ::> GB ::> A. Next we prove item (ii). If b € Bn then

for all k Gb c

€

:N

with k

>.·

n.

Now one easily notes that

n (bWk)w;1 S bG k>n

Gb c bG.

and so

Similarly Gb ::> bG and we have Gb = bG for all b € B. For any g € G and b € B, setting x := gb, we get g- 1 (Gx) • Ggb • Gb .. bG

and

so

Cx • gbG = xG.

By compactness of G, we conclude that Gx X €

xG, for all x

€

GB (i.e. for all

A).

Finally, we prove item (iii). Since B x :N is countable, by Urysohn's lemma we can find a sequence {gn} of continuous functions from AG into [o,l] such that, corresponding to each (b,m) € B x lN there is an n € lN for which Sn(b) .. o,

gn(x) • 0

for all

x

€

AG \ Wn-1 (Wn b) •

[Note that w~ 1 (Wnb) is open, by Theorem 2.6(iv) .J For each n fn(x) :• 2-n(x*v(Sn))

56

for all

x

€

AG.

€

:N,

let

... Let p(x,y) :=

~ If (x)-f (y)l for all x,y

n=l

n

n

continuous semimetric on GA.

AG, and note that pis a

€

Now one can easily show that every f

€

C(GA),

such that x*w(f) = f(x) (x € GA) is continuous with respect to the semimetril p. Since GA is compact, item (iii) now follows. 4. ACTIONS ON NORMED LINEAR SPACES We will soon define the action of S on a normed linear space

A. We are

interested in establishing cases where we have weak and norm continuity of such actions.

Our results provide a unified approach to various function

theoretic results involving uniformly continuous functions on S (as will be seen in Chapter 4) and measure theoretic results involving Ma (S) (as will be seen in Chapter 5). Let A be a normed linear space. 4.1 DEFINITIONS. say S has a We "'left action (or anti-action) on A i f there is a map (S ..IJ) -+ SOli of S x A into A such that xyop .. xo(yop) or (xyop = yo(xop), respectively), II sop II < liP II and the map p -+ SOl! of A into A is linear, We also assume that lop .. ll for all p € A. for all s,x,y € S and p € A. If such an action or anti-action exists we then say p € A has a norm (or weakly) separable "'left o~bit ove~ U ~ S if there exists a countable subset C of U such that {xop : x € C} is norm (or weakly, respectively) dense in {xop : x £ U}. Occasionally we shall simply write "separable" in place of "norm separable". [we shall also assume these definitions for general topological semigroups s.]

4.2 OBSERVATION.

We shall formulate our results in terms of actions and

remark that these results are also true for anti-actions as one can easily note that a left action for the semigroup (S,.) is a left anti-action for the semigroup (H,*) where H = S and x*y := y.x for all x,y

Let S have a "'left action on A, Then the following items are equivalent. 4.3 LEMMA.

(i) n has a weakly separable "'left

(ii)

n has a norm separable "'left

U

be a subset of

o~bit ove~

o~bit ove~

€

S. S

and

ll €

A.

U;

u.

57

Proof. Evidently (ii) implies (i). We now assume that (i) holds and prove (ii). Let C be a countable subset of U such that {xon : x e C} is weakly dense in {xon : X E U} and let co(T) denote the norm-closure of the convex hull of T := {xon : X E C}. Then co(T) is norm separable. Now {xon : x e U}

S weak-cl({xon

: x e C})

S weak-cl (c'O(T))

• co(T)

by the Hahn-Banach Theorem. So the norm separability of co(T) implies that of {xon : x e U} and we have item (ii) proved.

4.4 THEOREM. Let 5 have a te~ action on A and n e A be such that the set {s e 5 : II son-yon II ~ e:} is ctosed fozo at"L y e 5 and £ > o. Then item (i) impties item (ii) and item (ii) is equivalent to item (iii), ~hezoe: (i) n has a

sepazoable teft ozobit ovezo a neighbouzohood U of l; (ii) the~ s ~ so(xon) of 5 into A is no~ continuous~ fozo alt x e s 1 ; (iii) xon has a weakly sepa~ble left orbit ovezo some neighbouzohood U of 1 fozo at"L x e s 1 • ~eakty

Proof. Suppose item (i) holds. To prove (ii) it is sufficient to show that the maps~ so(xon) is continuous at 1, for all x e s1 • For then ff any x e s 1 is fixed and £ > 0 given, we have W := {s e 5 : llso(xon) - xonll < £} a neighbourhood of 1. Now if (s a ) is a net converging to s in 5, there is w e W n s 1 such that (sa) S (sW)w-l eventually, by Theorem 2.6(i). So we can find a net (w: c W such that, a eventually saw

= swa.

Hence, eventually

II s a o(xon)

- so(xon) II -<

II s a o(xon) +

<

58

- s a o(woxon) II

II so(wa oxon)

2£,

- so(xon) II

and item (ii) would follow.

X

Now suppose (ua) is a net converging to 1 in S, let £ > 0 be given and £ Sl be fixed. Then V := S-lx n U is a neighbourhood of 1, by the

definition of s 1 • So item (i) together with Lemma 4.3 imply that there is a sequence {s } c U such that n

-

CD

V

= u

(O(sn) n V)

where

O(sn) :• {y

£

S

n•l By the Baire category theorem (see e.g. Kelley [65]) we have

+ for

int(O(sk) n V) I

0 := int(O(sk) n V).

w

£

0 n

s1

yw '"'

and y

£

some positive integer k.

s1

Since

For our convenience let

is dense inS and 0

S S-1 x,

we can find

S such that (1)

X

-1 Recalling Theorem 2.6(i), we have that (u y) c (yO)w eventually. a

-

So we

can find a net (za) c 0 such that, eventually uayw

= yza.

(2)

Hence, eventually

II ua o(xon)

- (xon)

II

II uaxon - xon II II uaywon - ywon II • II yzaon - ywon II llyo(zaon) - yo(won) < <

by {1) by (2)

II

llzaon - wonll 2£

since (za) C 0 and w

£

0.

As remarked before, this completes our deduction of item (ii) from (i). A standard argument easily shows that (ii) implies (iii) : for if U is s £ any compact neighbourhood of 1, then (ii) implies that {so(xon) is a compact subset of the (metric) space A for all x £ s 1 • On the other hand suppose item (iii) holds and let £ > 0 be given.

U}

It is

evident that, since the set {s £ S : llson - yxonll ~ £} is closed, we ha11e the set {s £ S : II so(xon) - yo(xon) II ~ d closed, for all y £ S and x £ s 1

59

Hence the maps+ soyo(xon) of S into A is norm continuous (x,y € s 1), since (i) implies (ii). Now yo(xon) = yxon and s1s 1 = s1 , by Corollary 2.5. So item (ii) holds. This completes our proof.

Let G be a compact group bJith a left action on A and n 4.5 LEMMA. Then the follobJing items aPe equivalent:

£

A.

(i) the map x + xon of G into A is ~eakly continuous; (ii) the map x + xon of G into A is no:rm continuous. Proof. Evidently (ii) implies (i). Now suppose (i) holds and let M!(G,A) :• {p € A : the map x + xop of G into A is norm continuous}. Because x + xon is weakly continuous, the weak integral von := xondv(x) exists for each v € M(G). (For matters to do with weak integrals (and vector integration) we refer the reader to Diestel and Uhl [27] .) Moreover, since the convex-hull of the weakly compact set {xon : x € G} is weakly relatively compact, we have von € A. The map v + von of M(G) into A is clearly (norm to norm) continuous. Now for each v € Ma(G) we have x + X.v a norm continuous map of G into M(G), so we conclude the composite

f

is norm continuous into A. Now, by definition of the weak integral, we have h(von) •

J h(xon)dv(x)

(v

£

M(G) and h

€

A*).

Since x + h(xon) is continuous (h € A*) and G is compact, we easily see that v + von is continuous from M(G) with the weak* topology to A with the weak topology. Hence, since Mn1 (G,A) is (norm and hence) weakly closed and Ma(G) is weak* dense in M(G), we conclude that r. = lon is in M!(G,A) and our lemma is proved.

4.6 THEOREM. If S has a left action on A and n equivalent:

£

A the follobJing items

aPe

(i) the map s + son of (ii) the map s + son of

60

s into s into

A is ~eakly continuous; A is no:rm continuous.

Proof.

It is clearly sufficient to prove that (i) implies (ii).

suppose n show that

A satisfies (i).

E

O(y) := {s is closed.

E S

So let x

O(yon) :• {"

E

Let£> 0 be given andy

E

So

S be fixed.

Wen~

llson - yonll .,:: £} E

O(y).

Setting

II" -

A :

yonll

< £},

we note that the Hahn-Banach theorem implies that xon

E

weak-cl(O(yon)) = norm-cl(O(yon)) = O(yon).

Consequently llxon- yonll .,:: £ and·x

E

O(y).

So O(y) • O(y) and O(y) is

closed. 1

Next we note that Mn(S,A) :=

{\I E

A : x

+

xo\1 is a norm continuous map

of S into A} is norm and so weakly closed in A. Let A be a compact neighbourhood of 1 and suppose (for the moment) that 1 is ~-isolated.

Then (the right handed version of) Proposition 3.8(i) say!

that A=

BG

for some countable B ~A and compact subgroup G of S. that {xon : x

E

By Lemma 4.5 we have

G} is norm compact and so we can find a countable subset D

of G such that {xon : x

E

D} is norm dense in {xon : x

E

G}.

Now for each u in A we have that uon

E

weak-cl({bxon : b

B and x

E

E

G})

since A

- weak-cl({bo(xon)

b

E

B and x

E

G})

= weak-cl({bo(don)

b

E

B and d

E

D}) since n

weak-cl({xon : x

E

= BG E

M!(G,A)

BD}).

Since BD is countable we thus have that n has a weakly separable left orbit over A. 1 So by Theorem 4.4 we have xon E Mn(S,A) for all x E dense in S and M!(s,A) is weakly closed, we thus have

n

E

weak-cl({xon : x

E

s1 •

Since

s1

is

1 s1 }) S Mn(S,A).

Thus our theorem follows for the case where 1 is

~-isolated.

61

Finally we suppose that 1 is not necessarily 6-isolated. Let U be any fixed compact neighbourhood of 1 and x £ s 1 • Lemma 3.4 says that we can find a 6-isolated idempotent e in S such that Ux c eSe. Thus eSe is a stip with a 6-isolated identity e; so it follows that-n £ M!(eSe,A) and hence that {yon : y £ Ux} is a norm compact subset of A. Hence {uo(xon) : u £ U} contains a countable norm (and so weakly) dense subset. By Theorem 4.4, we thus have yo(xon) £ M1 (s,A) for ally£ s 1 • Since x was arbitrarily chosen n t in s 1 and s 1 • s 1s 1 we have son£ Mn(S,A) for all s £ s 1 • Now 51 = S, so item (i) and the weak-closure of M!(s,A) imply that n £ weak-cl({son : s £ s 1 }) c M1 (s,A). n Thus (ii) holds and our proof is complete.

4.7 REMARK. Recalling the proof of Lemma 4.5 and the inner regularity of Radon measures, the reader should now be in a position to prove the following generalization of Lemma 2.1.6(ii) : Let S be any topological semig~up acting on a norn~ed linea:!' space A such that the map x -+ xon of S into A is weakly continuous fozo some n £ A. Then fozo all + £ A* we have +(von) =

whezoe von :•

62

J +(xon)dv(x)

J xondv(x),

fozo all v £ M(S).

4 Algebras of functions

Various results on algebras of functions on a topological semigroup S can be found in the literature.

These include results on the so called algebras

of weakly almost periodic functions, uniformly continuous functions and other subalgebras of C(S).

It is not our intention in this chapter to give

a complete study of such function algebras;

instead, we extract some of

those results whose success is largely due to the existence of absolutely continuous measures or a related property on the underlying semigroup.

Thus

to a large extent this chapter deals with some applications of our studies in Chapters 2 and 3. 1. UNIFORMLY CONTINUOUS FUNCTIONS First we collect together some definitions and notations. 1.1 DEFINITIONS.

Let S be a topological semigroup.

A function f in C(S) is said to be left uniformly (or left weakly

uniformly) continuous if the mapping x

~

xf of S into C(S) is norm (or

weakly respectively) continuous. We denote the space of all left uniformly continuous functions on S by LUC(S) and that of left weakly uniformly continuous functions by Similarly one defines the spaces RUC(S) and

Rl~C(S)

Ll~C(S).

of right uniformly

(respectively) continuous functions on S. We say a function is uniformly continuous on S if it belongs to the space UC(S) := LUC(S) n RUC(S), and weakly uniformly continuous if it belongs to the space WUC(S) := LWUC(S) n RWUC(S). A function f in C(S) is said to be weakly almost periodic if the set { f : x X

€

S} is relatively weakly compact.

We denote the set of all weakly

almost periodic functions on S by WAP(S). Although we have defined weakly almost periodicity of a function in terms of its left translates an equivalent definition in terms of right translates is also acceptable.

This follows easily from the following result which is

an immediate consequence of Grothendieck's Theorem (see Appendix B.7). 63

LetS be a topological semigroup and items are equivalent:

1.2 PROPOSITION. foll~ing

f

Then the

C(S).

£

(i) the set {X f : x £ S} is relatively weakly compact; (ii) ~henever {xn} and {ym} are sequences in S the intersection of the closures of the sets {f(xnm y ) : n < m} and {f(x y ) : n > m} is non-empty. nm (iii) the set {fx : x

£

S} is relatively

~eakly

compact.

As an application of Theorem 3.4.4 we have the following characterization of uniformly continuous functions in terms of separable orbits.

First we

note that a topological semigroup S has a natural left anti-action on C(S) give by the map (s,f)

~sf

of S x C(S) into C(S).

Lets be a stip and f (ii) and (iii) are equivalent~ ~here

1.3 THEOREM.

(i) f has a

£

With this in mind we have

Then item

C(S).

(i)

implies

(ii)

and

left orbit over a neighbourhood of 1; (ii) xf is left unifol'171ly continuous~ for all x £ s1 ; (iii) xf has a ~eakly sep~le left orbit over a neighbourhood of 1~ for all x £ s1 . Proof.

Let y

~eakly sepa~le

S and £ > 0 be given.

£

Then this Theorem will follow from

Theorem 3.4.4 i f we can show that the set {s £ S : llsf - yfll S ~ d closed. So suppose (s a ) is a net converging to s in S with

lis a f

-

lll s ~

£.

Hence lf(s a z) - f(yz)l <£for all z

lf(sz)- f(yz)l ~£for all z

II sf

-

/II

£

S.

F.

is

Sand so

Thus

S := sup{ If(sz) - f(yz)

I

z

£

S} ~

£

and our result follows.

1.4 EXAMPLE.

We give an example of a stip S and a function f

£

C(S) with a

separable left orbit over a neighbourhood of 1 but is such that f is not left uniformly continuous.

We consider the following example of a stip S

which is also the foundation of Ma(S). Let S '"' {- 2 n1 : n with the operation 64

£

1

1N} u {0} u { - -

2n+l

n

£

1N}

xy = yx = x

if

lxl ~

IYI•

for all x,y

E

S;

and the topology of S coincides with the restriction of the line topology on {- 2n1 : n e

1N}

1 u {0} while its restriction on {0} u {2ii+f : n e

x e Ma (S)

discrete. Since for all x e S\{0} we have xS finite,

1N}

is

by

Exercise 2.3.10(a)(i). Thus Sis the foundation of M (S) with identity a element 0. Since S is countable, the function f e C(S) given by 1 for x > 0 f(x) := { 0 for x ~ 0, (x e S) has a separable left orbit over S.

II xn f-

o

fll

s

+1

andso

! then xn + 0 while

But if xn := - 2

f.LUC(S).

1.5. For every topological semigroup S one easily notes that So for a stip S, taking A := C(S) in Theorem 3.4.6, we have THEOREM.

If S is a

(i) LUC(S) (ii) WAP(S)

stip~

WAP(S~WUC(S).

we have that

LWUC(S);

S UC(S).

1.6. The preceding results are taken from Dzinotyiweyi ([33] and [34]). When S i~ a locally compact group, Theorem 1.5(ii) has been proved in a different way by Burckel [13]. We now turn to the situation where the underlying topological semigroup Sis assumed to be the foundation of Ma(S). Hence we remind the reader that the term "foundation semigroup" mentioned hereafter is as defined in item 2.3.3. The results given in the remainder of this section are taken from Dzinotyiweyi and Milnes [38] •

any topoLogicaL semigroup S wheneve~ we taLk of a jUnction f in m(S) as having a (weakLy) separabLe Le~ o~bit~ this wiLL be done in te~s of the Le~ anti-action of S on m(S) given by the mapping (s,f) + sf of S x m(S) into m(S), fo~ aLL s e Sand f e m(S). FUrthe~~ f is said to be measurabLe i f f is p~asurabLe~ fo~ aLL P e Ma(S). Fo~

65

1.7. The following remarks will enable us to simplify the proof of our next lemma. Let S be a foundation semigroup with identity element 1. Let B be the space of all bounded linear operators from Ma(S) into LUC(S) with the norm operator topology and note that S has a left action on B given by the mapping (s,P) + soP of S x B into B. Here soP(v) := P(s*v)

:= {P

Let B

€

for all" in Ma (S).

B : the mapping s

note :hat the integral voP := P

€

Ba (see e.g.·

(v,P)

+

II s

voP.

[27]).

Given e

+

soP of S into B is continuous} and hence

f soPdv(s)

exists for all v in Ma(S) and

Thus Ba is an Ma(S)-module under the map >

II soP - P II

0 and P in Ba we have that

d is an open neighbourhood of 1· and hence, for v in Ma(S) with v(W) = llvll = 1, we have

W :=

S :

€

llvoP - Pll

=

<

II f (soP-P)dv(s) II

.: _ f

llsoP-PII dv(s)

< e.

Thus the Ma(S)-module Ba has a bounded approximate identity, so by Cohen's factorization theorem we have that

(The reader can find details on Cohen's Factorization Theorem for the case of Banach Modules in e.g. Bonsall and Duncan [11].)

Let S be a foundation semigroup with identity element 1 and f € m(S) a function with a separable left orbit over a neighbourhood of 1. Iff isMa (S)-measurable, then x f is equal almost everywhere (i.e. a.e. with respect to every measure in Ma(S)) to a function in LUC(S), for all x in s 1 • 1.8 LEMMA.

Proof.

For every "

€

Ma(S) and Ma(S)-measurable h in m(S) we recall that

the functions voh and hov on S are given by voh(s) :=

66

f

h(ys)dv(y)

and

hov(s) :=

J h(sy)dv(y)

(s

€

S),

Now one notes that, for all x and y in S,

and so voh is in LUC(S) since the map r v of S into M(S) is norm continuous by Theorem 2.2.7. We now show that xoF

£

Ba for all x

£

s1 ,

where F

£

B is given by

F(v) :• vof It is trivial to note that, for all x and y in S, we have

II xoF

- yoF II -<

II Xf

- y f II 8

and so it follows that F has a separable left orbit over a neighbourhood of 1 Further, given £ > 0 we note that O(x) :• {s £ S : llsoF - xoFII < £} is closed, for all x in S. For if v £ Ma(S), then indeed

is closed and clearly

Hence, by Theorem 3.4.4, we have that xoF

£

Ba, for all x in

s1 •

Next we fix x £ s 1 • Recalling that Ba = Ma(S) o Ba• we can find p £ Ma(S) and H £ Ba such that xoF • poH. Then for every v in Ma(S) we have that

and (poH)(v) :•

J (yoH)(v)dp(y) • f H(y*v)dp(y) • H(p*v) • H(p)ov.

[The reader may refer to Diestel and Uhl (27] for details on the vector integral.] So, noting that h :• H(p) is in LUC(S), we have

67

voxf = hov

for all v in M (S). a

In particular f<xf(s) - h(s))dv(s) • voxf(l) - hov(l) • 0 By Proposition 1.3.3 we conclude that xf = h almost everywhere.

Let S be a foundation semigroup '1.11ith identity eLement 1, f £ m(S) and h £ C(S) such that f = h almost everYfJ}here and f has a separabl.e "left orbit over a neighbourhood U of 1. Then xf is in LUC(S) for a1-l. x in s 1 • 1.9 LEMMA.

Proof. Let x £ s 1 be fixed. follow, by Lemma 1.8.

We now show that xf = xh• and the result will

Suppose on the contrary there exists s 0

£

S such that

Let W be a neighbourhood of 1 with WW c U. By Theorems 3.2.2 and 3.2.6(iii) -1 we can find x 1 £ xs 1 n W n s 1 and hence x 2 £ s 1 with x 1x 2 • x. Since x 1 £ s 1 n W, the definition of s 1 implies that we can find a neighbourhood V of 1 with V c l~-1 x 1 n l~. By the left translation invariance of Ma(S), the set N(s) := {y

£

S : sf(y)

~

sh(y)} is Ma(S)-negligible.

The given hypothesis implies existence of a sequence {s } n

{snf : n

£

Ill}

is dense in {sf : s

£

U}.

~

-

U such that

Now N :=- n~l N(sn) is Ma(S)-

negligible and soH := S\N is dense in S, since S is a foundation semigroup. -1

By (the right handed version of) Theorem 3.2.6(i) we have t 1 (Vx 2 s 0 ) a neighbourhood of x 2 s 0 , for some t 1 £ V n s 1 • Since H is dense, we can find

Thus t 1 s • v 1x 2 s 0 for some v 1 in V. wv 1 = x 1 • Hence

68

Since V c

l~

-1

x 1 , we can find w in

l~

with

Setting t := wt 1 we have t

€

WW c U and

Now tf(s) := f(ts) = f(xs 0 )

~

f(xs 0 ) = h(ts) • th(s).

Since H\N(t) is dense in S, we can find a net (z ) c H\N(t) such that z a a Then

Hence tf. C(H). can find a c5

>

~

s.

However the functions

fare continuous on Hand so we sn 0 such that II tf - s fll H ~ c5 for all n in lN. Hence n

II tf

- s f n

II

5

~

II tf

- s f n

II H ~

c5 > 0,

for all n

This inequality contradicts the fact that {snf : n {sf : s

€

U}.

€

lN}

€

lN.

is dense in

By this conflict our lemma follows.

We recall that for a stip Sand f

€

C(S), Theorem 1.3 says that f is

"very close" to being left uniformly continuous as soon as f has a weakly separable left orbit over a neighbourhood of 1. Our proof there is entirely independent of measure theory and relies more on topological techniques.

If

in addition S is a foundation semigroup then a more general result can be achieved via measure theoretic techniques. Theorem.

This is the message of our next

LetS be a foundation semi~up with identity element 1 and a fUnation with a weakly left sepaPable oPbit oveP a neighbouPhood of 1. If f is Ma (S) -measumble then sf is left uniformly aontinuous, foP all x in s 1 • 1.10 THEOREM. f € m(S)

Proof.

Lemma 3.4.3 says that f has a separable left orbit over some

neighbourhood of 1. LUC(S), for all x,y conclude that xf

€

Lemmas 1.8 and 1.9 then imply that xy f • y ( x f) is in s 1 • Since s1s 1 • s1 (see Corollary 3.2.5), we LUC(S), for all x € s 1 • €

69

If we visualize Theorem 1.10 as the corresponding measure theoretic generalization of Theorem 1.3, then we may take the following result to be the corresponding measure theoretic generalization of Theorem 3.4.4.

Let S be a foundation semigroup with identity element 1 and let S have a left anti-action on a normed linear space A. Let n e A have a ~eakly separable left orbit over a neighbourhood of 1 and suppose the functions s ~ ~(son) of S into Care Ma(S)-measurable, for all ~ e A*. Then s ~ so(xon) is a norm continuous rmpping of S into A for all x e s1 . 1.11 COROLLARY.

Proof.

Fix

~

e A* and consider the function f : s ~ ~(son) of S into C.

By Lemma 3.4.3 we may assume that n has a (norm) separable left orbit over a neighbourhood of 1.

Now for all

~

e C(S)* and x,y e S we have that

and so f has a weakly separable left orbit over a neighbourhood of 1. Theorem 1.10, we have that xf e LUC(S), for all x in xf e C(S) and so the mapping s Sl,

~

s1 •

By

In particular

so(xon) of S into A is weakly continuous,

Recalling Theorem 3.4.6 our result follows.

for all

X E

1.12.

In a general sense a function f e m(S) is said to be almost periodic

on the semigroup S if the set { f : x X

E

S} is relatively norm compact.

A

well known result of von Neumann says that a Haar measurable almost periodic function on a locally compact topological group is continuous.

Now if S is

a locally compact topological group and xf e C(S) for some x in S then f

C(S), and a function is Haar measurable if and only if it is Ma(S) -

E

measurable.

Noting that every almost periodic function on a topological

semigroup S has a separable left orbit over S, by Theorem 1.10 we obtain (in an elementary manner) the following generalization of von Neumann's result. (Davis

[24]

also proved von Neumann's result, but the proof is rather long

and technical.)

Let S be a foundation semigroup with identity element 1 and f e m(S) an almost periodic function.

COROLLARY.

Iff is Ma(S)-measurable, then xf is a continuous almost periodic function. (i)

70

If S is a group and atmost periodic jUnction. (ii)

1. 13

REMARK.

f

is Haazo measumbte, then

f is

a continuous

Since there are non-continuous characters on some locally

compact topological groups and characters are almost periodic functions, the measurability condition cannot be completely dropped in Corollary 1.12 and hence in Theorem 1.10.

[see e.g. Hewitt and Ross [59, page 405] for

the existence of discontinuous characters on the additive group 1R~ 1.14.

We recall that on a locally compact topological space countable

unions of meager sets are meager.

Now the reader can easily note that;

if

in the proof of LeiiDJia 1.9, we replace "Ma(S)-negligible" by "meager", use Theorem 1.3 in place of LeiiDJia 1.8 and recall LeiiDJia 3.4.3, we have the following extension of Theorem 1.3.

(We leave the details as an exercise

for the reader.)

Let S be a stip and f E m(S) a jUnction that has a teft separable orbit over a neighbourhood of 1 and such that the set {s E S : f(s) ~ h(s)} is meager, for some hE C(S). Then xf E LUC(S), for aU X E Sl •

PROPOSITION. ~eakty

2. INVARIANT MEANS

Throughout this seJtion let S be any topological semigroup with an identity element 1 and such that s .is a k-space coinciding ~ith the foundation of Ma(S) unless othel'wise specified. In particular, recalling Theorem 2.1.8 and Corollary 2.2.7 we have that S is

C-distinguished and Ma(S)

2.1 TERMINOLOGIES.

~

Mn(S).

Let A be a closed subspace of M(S)* such that xeh E A

for all x E S and h E A, and e E A where e is the functional which maps to

~(S)

for

all~

~

E M(S).

An element m of A* is a mean on A if llmll = m(e) = 1

and

m(f) ~ 0

for every non-negative functional f in A. A left invariant mean (LIM) on A is a mean m E A* such that

71

m(ieh)

= m(h)

for all

x £ S and h £ A.

H of M(S), let P(H) := {p £ H : p

For any subset

~ 0 and

!lull

= 1}. Now

if A is such that peh £ A for all p £ H and hE A, a P(H) -Left invaY'iant mean

(P(H) - LIM) on A is a mean m £ A* such that m(peh)

= m(h)

for all

p £ P(H)

and h £ A.

In particular a P(M(S)) - LIM on A is called a topoLogicaL left invaY'iant

mean (TLIM).

(Trivially a TLIM is always a LIM.)

If A is a closed translation invariant subspace of m(S) containing all constant functions, then we naturally take a LIM (or (TLIM) on A to be a mean min A* such that m(xf)

= m(f)

(or m(pof)

= m(f),

respectively) for

all x £ S, f £A and p £ P(M(S)). Let H be any locally compact topological semigroup.

AmenabiLity of H

has been defined to mean the existence of a LIM on (a) LUC(H) by e.g. Namioka [74] and Jenkins [63];

(b) MB(H) -the measurable functions in m(H)

where His a subsemigroup of a group, by e.g. Lau [71]; e.g. Wong [104]. In the case where H is a group, Greenleaf

[53]

and (c) M(H)* by

showed that all such

definitions are equivalent and also are equivalent to related definitions for a TLIM. We extend Greenleaf's result to topological semigroups in Theorem 2.4.

Our Theorem could also be deduced from Theorems 6.6 and 6.7 of

Paterson

[so]

if S is assumed to be locally compact.

The actual presentation

given here is taken from Dzinotyiweyi [31]. For a comprehensive account on the subject of invariant means on locally compact topological groups we refer the reader to Greenleaf

[53}.

2.2 LEMMA. Let f £A and p £ M(S),whePe A is any of the spaces UC(S), LUC(S), RUC(S), C(S) and MB(S) := {g £ m(S) : g is a Borel function}. Then (i) uof

£

A;

(ii) if p,v £ M (S) and h £ M(S)*, then poh £ LUC(S) and uohov £ UC(S). a

Proof.

(i) Note that for all x,y £ S we have uofx(y)

llyll

72

=

lluofx- uofyll 5 ~!lull

= (uof)x(y)

llfx- fyll·

and so

Hence item (i) holds for A= RUC(S). Also item (i) holds for A = C(S) by Lemma 1.3.4. Next we show that f £ LUC(S) implies that pof £ LUC(S). Evidently LUC(S) is norm and hence weakly closed. If on the contrary pof. LUC(S), the Hahn-Banach Theorem requires the existence of a + £ LUC(S)* such that +(pof) I 0 and+= 0 on LUC(S). Then recalling Remark 3.4.7 we would get the ridiculous situation

oI

+(pof) =

J+<xof)dp(x)

=

o.

By this conflict we conclude that pof £ LUC(S). Consequently item (i) also follows for A= UC(S), and it remains to show that f £ MB(S) implies pof £ MB(S). To the latter end, we first note that we may assume p to be a positive measure without losing generality. Now for any closed set F C S we have that the function poxF is upper semicontinuous, by Lemma 1.4.l(ii), so poxF is Borel measurable. Next if {F } is an increasing (or decreasing) sequence of Borel sets with n

poxF measurable (n £ 1N) then, for all s £ S, n

POXF (s) :• n co

f XFn (xs)dp(x)

-+

lJOXF(s),

co

where F := u Fn (or F := n Fn• respectively). Recalling the monotone n=l n=l class lemma and the fact that closed subsets generate the Borel subsets of S we have poxB £ MB(S) for every Borel set B C S. Finally if {fn} is a bounded sequence of simple Borel functions converging pointwise to f, then lJofn(s) + pof(s) for all s £Sand so pof £ MB(S). (ii) For all x,y £ S we have llx(poh) - y(poh) II S = sup{ lh(p*xs) - h(p*ys) I ~

II h II

s

£

S}

llp*x - P*Y II •

Since M (S) c M (S), we have poh £ LUC(S). a - n

Noting that

pohov = (peh)ov = po(hev) and peh, hev £ M(S)*, it follows that pohov £ UC(S). 73

The argument involving MB(S) in our Lemma is taken from Paterson 2.3 LEMMA.

Let m be a LIM on . UC(S).

Lao].

Then

(i) m is a TLIM on UC(S); (ii) m(pohov) = m(nohov) Proof.

For any

~· E

fo~

P(M(S)) and f

m(~of) = J m(xof)d~(x) •

aLL v,p,n

m(f)~(S)

E

E

P(Ma(S)) and h

E

M(S)*.

UC(S) we have

by Remark 3.4.7

• m(f);

and so item (i) follows. We now prove item (ii). By item 2.2.11 we may take (pa) to be a bounded (norm) approximate identity forM (S). Then a

lm<~ohov) - m(pa*~ohov)

I

~ sup{ llhll

ll~*x*v - Pa*~*x*vll

~ llhll u~-pa*~ll-+

X E S}

0,

(with notation as stated in the Lemma). By (i), we have m(pa*~ohov)

•

m(~o(paohov)) =

m(no(paohov)) • m(pa*nohov)

and so lm(~ohov)- m(nohov)l < lm(~ohov) - m(p *~ohov)

a

I

+ lm(p a *nohov) - m(nohov>l

-+ 0 and our result follows.

Let A1 and A2 be any of the spaces: UC(S), LUC(S), RUC(S), C(S), MB(S), Ma(S)* and M(S)*. Then the foLLowing items ~e eq~ivaLent: 2.4 THEOREM.

(i)

The~e ~sts

a P(Ma(S)) - LIM on A1 ;

(ii)

the~e

exists a LIM on A2 ;

(iii)

the~e

exists a TLIM on A2 •

Proof. We note that UC(S) ~ A1 £ M(S)* for all choices of A1 • So the existence of any of the three types of left invariant mean on M(S)* implies (by restriction) the existence of the same type of left invariant mean on A1 • Now let m be a P(Ma(S)) -LIM on UC(S). 74

Then, with the above remarks in

mind and recalling that a LIM on UC(S) is a TLIM, our Theorem will follow if we can show that there exists a TLIM on M(S)*. To this end we fix measures n and v in P(Ma(S)) and define the function M on M(S)* by M(h) := m(nohov) Since nohov for any

~ E

E

for all

h

M(S)*.

UC(S), we evidently have that M is a mean on M(S)*.

P(M(S)) we have that

M(~oh)

E

~*n E

Further,

P(Ma(S)) and so

:= m(no(~oh)ov) = m(~*nohov)

= m(nohov) = M(h).

by Lemma 2.3(ii)

This completes our proof. 2.5 SOME OPEN PROBLEMS.

Of course the success of Theorem 2.4 is pivoted

around the object Ma(S).

It is natural to ask what the situation would be

when existence of Ma(S) is not assumed. For instance in Chapter 3 we studied an important class of semigroups S which behave as if they were foundation semigroups though the latter is still unknown.

The following

problem is therefore reasonable. PROBLEM A.

Let S be a stip.

We conjecture that

(i) there is a LIM on C(S) if UC(S) has a LIM; (ii) there is a TLIM on C(S) if there exists a LIM on C(S). Should the question whether every stip is a foundation semigroup turn out to be answered affirmatively then Problem A would follow.

However it would

still be interesting if Problem A could be settled without the use of Ma(S). The preceeding remarks invoke another question.

As stated before, our

Theorem 2.4 is a generalization of a result well known for groups - see e.g. Greenleaf [53, Theorem 2.2.1]. Now for a topological group G we know that G is locally compact if and only if Ma(G) is non-zero.

Hence considering

the situation where Ma(G) is zero, we have the following problem. PROBLEM B. Let G be any (non-locally compact) topological group. We do not know whether the existence of a LIM on C(G) implies the existence of a TLIM on C(G).

Further, we do not know whether there is a LIM on C(G) as 75

soon as there is a LIM on UC(G). First we warn the reader that, for the remainder of this

2. 6 TERMINOLOGY.

section, S is not necessarily as stated at the beginning of the section. Let S be any locally compact topological semigroup and assume the notations introduced in item 2.1 for this S. A net or sequence (pa) c P(Ma(S)) is said to be conve~ent

to topotogicat inuaPiance if v*pa - Pa

~eakty

+

(or

st~ngty)

0 and Pa*v - Pa + 0

weakly (or strongly, respectively) in Ma(S) for all v

€

Ma(S).

Let IM(WUC(S)) denote the set of (left and right) invariant means on WUC(S). We now commence an argument aimed at estimating the size of IM(WUC(S)).

Let S be a tocatty compact topotogicat then

2. 7 LEMMA.

Petativety

semi~up.

If S is not

neo-compact~

m(f) • 0 foP att f

€

C (S) and 0

m

€

IM(WUC(S)).

Let f € C (S) be positive and put K := supp(f). Proof. Since S is not oo relatively neo-compact, we can choose a sequence {xn} in S such that X

n+

So if n

~

l

! -1 -1 -1 't K(x 1· Ku ••• ux K)

for all n

n

-1

k we have xn K n

sup.p(x f) n supp( n

~

-1

·~

K=

+ or,

€

1N •

equivalently,

+·

f) •

Consequently nm(f) • m(

xl

f + ••• +

xn

f)

<

llfll S

for all

n

€

:N,

and so m(f) "' 0. Since C (S) is dense inC (S), the remainder of our result follows 00 0 trivially. (Note that C (S) c l~C(S), by Lemma 1.3.4 and the fact that C0 (S)*

76

= M(S).)

0

-

Let S be a C-distinguished topoZogicaZ semigPOup. The~ is a net in P(Ma(S)) ~eakZy convergent to topoZogicaZ invariance if and onZy if the~e is a net in P(Ma{S)) st~ongZy convergent to topoZogicaZ invariance. 2.8 LEMMA.

Proof. Suppose (n a ) c P(Ma (S)) converges weakly to topological invariance. Setting M := M {S) we form the locally convex product space \loll

a

M := IT{Mv,p : (v,p) E P(Ma(S)) x P(Ma(S))} with the product of norm topologies and define the linear map L Ma(S) + M by

Appendix B.J says the weak topology on M coincides with the product of the weak topologies on the Mv,p 's. Since v*n a *ll - na + 0 weakly, for all v,p E P(Ma(S)), 0 E weak-closure L(P(Ma(S)). Since L(P(Ma(S))) is a convex subset of the locally convex space M, we have weak-closure L(P(Ma(S))) = strong-closure L(P(Ma(S))), by the Hahn-Banach Theorem. So there exists a net (pa) c P(Ma(S)) such that L(p 8) + 0 in M or, equivalently, such that

Hence

II v*p 8 - P8 II

<

II v*p 8 - v*v*p 8*vii

<

liPs - v*ps*vll + II (v*v)*Ps*" - P8 11

+

for all v E P(Ma(S)). Similarly 11Ps*v-p 8 11 lemma follows trivially.

II (v*v) *P 8*v - p 8 II

+

+

o

0, and the remainder of our

2.9 LEMMA. Let S be a ZocaZZy compact topoZogicaZ semig~p that is not FeZativeZy neo-compact and Zet {pn} be a sequence in P(Ma(S)) stPOngZy conve~gent to topoZogicaZ invariance and such that Tn := supp(pn) is compact~ fo~ aZZ n e m. Then given n 0 e mand e: > 0, ~e can find n > n 0 such that lln ((T1u ••• uTn ) n Tn ) 0 'Proof.

<

e: •

Suppose, on the contrary, there exists an n0 and e: pn((T1u ••• liTn ) n Tn ) -> e: 0

>

0 such that

for all n > n 0 •

77

Let f E C0 {S) be a positive function with f

1 on T1u ••• uTn

and note that 0

~

(f)

n

> £

-

for all

n

>

n • o

Let m be any weak*-cluster point of

{~

n

invariant mean on WUC(S) such that m(f) m(f) = 0.

} in WUC(S)* and note that m is an ~ £,

By Lemma 2.7, we must

hav~

This contradiction implies our result.

2.10 LEMMA. Let S be a a-compact locally compa~t topological semigroup (with Ma(S) non-zero) and let there be an invariant mean on l~C(S). Then there exists a sequence {pn) in P(Ma(S)) converging strongly to topological invariance and such that Kn := supp{p n ) is compact~ for aU n E ~.if any

one of the foUowing conditions holds: (a) S

is a foundation semigroup

~ith

an identity element.

(b) The centre of Fa(S) is not Ma(S)-negligible. Proof.

First we show that there exists a net in P(Ma(S)) weakly convergent

to topological invariance, if (a) or (b) holds. holds.

To this end, suppose (a)

Then there exists a topological (left and right) invariant mean m

on Ma{S)*, by Theorem 2.4. Now m E weak*-closure (P(Ma{S))) in Ma(S)**· Consequently, there exists a net (~ a ) in P(Ma (S)) such that ~ a (h) + m(h) for all h E Ma(S)*. In particular, for each v E P(Ma(S)) we have

for all h E Ma(S)*, Similarly ~ a *v - ~ a + 0 weakly. is weakly convergent to topological invariance,

Thus

(~

a ) C P(Ma (S))

Next suppose condition (b) holds. Then we can choose T E P(Ma{S)) such that supp(T) c Z(F (S)) • where Z(F (S)) denotes the centre of F (S). l-Ie then a a a have T*v =.v*T for all v E P(M (S)). Now if m is any invariant mean on a o WUC(S) we have that m0 is topologically invariant, by arguing as in Lemma 2.3(i).

Let (n) be a net in P(Ma {S)) such that na (f)+ mo {f), for all a f E WUC(S). Then, for any v E P(Ma(S)) and hE Ma(S)*, we have that

T*vohoT, TohoT E WUC(S) by Lemma 3.2;

78

hence, if

~a

:= T*na*T,

• na(T*vohoT) - na(TohoT) m0 (vo(TOhOT)) - m0 (TOhOT)

+

a

0.

Similarly h(pa*v - pa) + 0. Thus (pa) is weakly convergent to topological invariance. Now suppose either (a) or (b) holds. Then there exists a net (n 8) S P(Ma(S)) strongly convergent to topological invariance, by Lemma 2.8. Fix any A E P(Ma(S)) and set Ps :• A*ns*A. Note that (p 8) is also strongly convergent to topological invariance. As S is a-compact, we can choose an

.

increasing sequence of compact neighbourhoos

n1 S n2 S ...• such that

Noting that the maps X+ x*ps and y + Ps*Y of s into Ma(S) are u Dn. n=l norm continuous (see e.g. Corollary 2.2.4) we can choose a sequence S =

{ps ,P 8 •••• } trom the ps's such that 1 2 llx*ps *Y - Ps n

n

II

<

s!

for all x,y

E

Dn.

Choose compact sets Kn such that

Setting p

n

a standard technical argument shows that

Consequently

79

Now for any v,n E P(Ma(S)) with compact supports we have supp(v) u supp(n) C D for n larger than some n and hence - n o :=sup{ lv*p n *n(f)-p n (f)

~

I:

f

Co (S), llfll 5 -< 1}

E

sup{Jix*pn*y(f)-pn(f)ldv(x)dn(y):f

E

C0 (S), llfll 8

~

1}


{cf» E

(t"")* :

cf»

~ 0,

11+11 = 1 and +(g) = 0 for all g

such that g(k)

~

0 as k

~

E

t""

""} and c denote the

cardinal of continuum.

Let S be a a-compact 'locaZZ.y compact topo'logica'l semigroup such that S is not l'e'lative'ly neo-compact~ Ma (S) is non-zero and thel'e e:cists an inual'iant mean on WUC(S). Suppose eithel' 2 .11 THEOREM.

is a foundation semigroup IIlith identity eLement~ Ol' (b) the centl'e of Fa(S) is not Ma(S)-neg'ligib'le.

(a) S

Then thel'e exists a lineal' isomet'!'y T(F)

Proof.

T :

(t"") ~

S IM(WUC(S)) and so card(IM(WUC(S)))

WUC(S)* such that ~ 2c.

Note that we have the hypothesis of Lemma 2.10 met, and let {pn}

and {Kn} be as in Lemma 2.10.

Choose v

P(Ma(S)) with C := supp(v)

E

compact and note that the sequence {v*pn*v} also converges strongly to topological invariance. Observi11g that Tn := supp(v*pn*v) = CKnC is compact (n E :m) and recalling Lemma 2.9, there exists a subsequence v*p

80

~

{~}

*v (T \ (T u ••• uT )) > 1-2-k ~ nl ~-1

of {n} such that

for k

=

2,3, ••••

Let

'If

:

.

WUC(S) ..... R.

'lr(f)(k) : = p

~

To see that follows:

1r

be the linear mapping defined by

(f).

for all f e WUC{S) and k e

is onto, let g

..

E R.

be fixed.

1N.

We now define "' g

E R.

..

as

Let

~(1) := g(l) and g(k) := g(k)-

k-1 I:

i=l

g(i)JJk(F.)/lJ.(F.), fork= 2,3, ••• 1 1 1

(Indeed~ e .t"" since, by our definition of the Fk's we have 180>1 ~ 2llgll .. and on assuming ls(i>l ~ 2llgll .. fori= 1,2, ••• ,k-l we are led to k-1 k-1 . lg(k) I~ 11811 .. + 2 11811 .. i:l ~(Fi)/}.li (Fi) ~ 11811 .. (1+2 i:l 2-k/(1-2~ 1 ))

~2llgil ... >

Now the pairwise disjointness of members of the sequence {Fk}, that "' g e R... and 'IJk(Fk) h

. l,

>

:=

imply that the function

is a linear functional in M(S)*. Consequently vohov e WUC(S), by Lemma 3.2. n(vohov)

Now, for all k e

~.

we have

= p (vohov) r'k

v*p

*v(h)

..

~

'IJk(.E (g(i)/'IJi(Fi))xF.) 1=1 1 k-1 = g(k) + I: g(i)'IJk(F.)/lJ.(F.) = g(k). i=l 1 1 1 Thus

1r

maps the function vohov onto g and

1r

is onto.

Further, we clearly have 11811 ..

=

lln 11 .. = llvohovll 5 •

It follows that the dual map n* :

{R.00 )* -+

WUC(S)* is a linear isometry.

81

~•F

To see that ~*+

> 0

S IM(WUC(S)), let+£ F be fixed. ~*+(1)

and

Then, clearly

= +(1) = 1.

Now for any n £ P(Ma(S)) and f £ WUC(S) we have

:c p

~(nof-f)(k)

~

(nof-f)

Recalling the definition of ~*+(nof-f)

Similarly

=

= (n*P

-p )(f)+ ~~

o

ask+~.

F we have

o.

~*+(fon-f)

= 0

and so

~*+

£ IM(WUC(S)).

Taking SlN to be the Stone-Cech compactification of lN, we have SlN\lN SF.

Since card(SlN\:fi) = 2c and~· is an isometry, we thus get

card(IM(WUC(S))) ~ card(f) > 2c. Theorem is proved.

SoT:= ~* is the required map and our

Let S be a non-FeZativeZy neo-compact~ a-compact~ ZocaZZy compact topoZogicaZ semigroup ~th Fespect to which the hypothesis of TheoFem 2.11 hoZds and Zet E(S) := {f £ WUC(S) : m(f) = a constant as m runs through IM(WUC(S))}. Then the quotient space WUC(S)fE(S) is nonsepaFabZe. 2.12 COROLLARY.

Proof.

Suppose, on the contrary, WUC(S)fE(S) is separable.

Then there

exists a countable set F := {fn : n £ lN} in WUC(S) such that E(S) + L(F) is dense in WUC(S), where L(F) denotes the linear span of F.

So each

p £ IM(WUC(S)) is determined by the sequence {p (f ) } and so n

card(IM(WUC(S))) 2.13 NOTES.

~c.

This contradicts Theorem 2.11.

Various results on the sizes of sets of invariant means can

be found in the literature : for discrete semigroups see e.g. Chou ([15] and [16]), -Granirer [51] and Klawe [68], and for locally compact topological groups see e.g. Chou ([16] and [17]) and Granirer [49]. Our Theorem 2.11 generalizes some of these results and our techniques follow closely those given in [17].

[49].

The proof of our Corollary 2.12 is similar to that mentioned for

groups in [18] •

82

For gorups, Corollary 2.12 was first proved by Granirer

3. THE SIZE OF THE DIFFERENCE WUC(S)\WAP(S) In this section, for a large class of topological semigroups

s,

we indicate

how one can use absolutely continuous measures to explicitly construct many functions in WUC(S)\WAP(S).

FoP convenience ~e assume that aZZ the spaces of measupes~ jUnctions (and jUnctionaZs) mentioned in this section aPe FeaZ-vaZued and PemaPk that the genePaUzation to the compZex-vaZued case can be achieved tPivici.Uy. First we prove a lemma which plays a pivotal role in the construction of functions in WUC(S)\WAP(S).

Let s be any topoZogicaZ semigPOup

~hich

is not PeZativeZy neoLet C := C0 u {1} D := D u{l} ~hePe 1 is an identity eZement of S (if thePe is one) OP an 0 adjoined isoZated identity eZement of s. Then theFe exist infinite sequences {xl'x2 , ••• }and {y1 ,y 2 , ••• } in S such that 3.1 LEMMA.

compact~

C0 and D0 any fixed compact subsets of S.

C-l(Cx y D)D-l n C-l(Cx.y.D)D-l n m

if any one of the (a) n < m

~

foZZo~ing

and i

=+

thPee conditions hoZds:

> j;

(b) n > m, i > j

and n

(c) n ~ m, i ~ j

and

Proof.

J

~

i;

m ~ j.

Our proof is by induction.

Suppose, by the inductive hypothesis,

we have finite sequences Xp := {x 1 ,x 2 , ••• ,xp} and Yp := {y 1 ,y 2 , ••• ,yp} inS such that the lemma holds for n,m,i,j in {1,2, ••• ,p}. For convenience, let p-1

and

Rp :=

u

m=l In terms of the latter notation, the conclusion of our lemma under item (a) for the finite sequences X and Y , is equivalent to p

p

(1)

83

We now establish the inductive step, that is choose xp+l and yp+l such that the lemma is valid for n,m,i,j in {1,2, ••• ,p+l}. Since S is not relatively neo-compact while both T :• C-l(C(C-lL D-l)D)(Y D)-l p

and

p

T' :~ C-l(C(C-l(CX Y D)D-l)D)(Y D)-l p p p are relatively neo-compact, we can choose xp+l in S\(TuT'). Now, that xp+ 1 ~ T is equivalent to C-l(Cx 1Y D)D-l n C-lL D-l • p+ p p

(2)

+

while that xp+l ~ T' is equivalent to (3)

Also the subsets

Q :• (CXp+l)-l (C(C-l(CXPYPD)D-l)D)D-l and Q' :• (CXp+l)-l (C(C-l(RpuCxp+lYpD)D-l)D)D-l

are relatively neo-compact, and so we can choose yp+l in S such that yp+l~ Q and yp+l ~ Q'. Equivalently, this is such that

+

(4)

n C-l(CX Y D)D-l •

(5)

n C-l(R u Cx 1Y D)D-l • p p+ p

p p

and

+

(respectively). Now for the finite sequences Xp+l and Yp+l" item (3) and the inductive hypothesis show that the lemma holds under condition (b), item (4) and the inductive hypothesis show that the lemma holds under condition (c), and to verify the lemma under condition (a) it is sufficient to establish item (1) with p+l in place of p. To the latter end, we note that the inductive hypothesis, items (2) and (5) imply that

84

C-lL

p+l

D-l n C-lR D-l • C-l(L u CX 1y 1D)D-l n C-l(R U Cx Y D)D-l p+l p p+ p+ p p+l p (C-lL D-l n C-l(R u Cx 1Y D)D-l) p p p+ p (C

-1

(CXp+lyp+lD)D

-1

-1

n C

(Rpu Cxp+lYpD)D

-1

)

= ••

Repeating the argument countably many times we get our lemma.

This

completes the proof. Remark.

We warn the reader that in general conditions (a), (b) and (c)

together, are weaker and not equivalent to the condition (d): (n,m)

~

(i,j).

It remains an open problem whether one can establish the conclusion of our lemma under (the stronger) condition (d). group S, this is not known to us.

In fact even for a topological

We strongly believe that if one can

supply an affirmative answer to this problem, for a locally compact topological group S, then property (E) mentioned by Ching Chou [18] may be dropped in most of the results of [18]. Next we introduce the role of absolutely continuous measures in the following lemma.

Let s be a c-distinguished topological semigPOup~ v and ~ measures in M(S) such that the maps X ..... v*x and X ... x*~ of s into M(S) a1'e weakly continuous. Then 3.2 LEMMA.

voho~ £

WUC(S) for all

h

£

M(S)*.

We first note that for all A £ M(S) and h £ M(S)*, vohoA £ C(S), Proof. since vohoA(x) = heA(v*x) and the map x + v*x is weakly continuous. To show that voho~ is in RWUC(S), for example, we take a t £ C(S)* and must show that the function x

+

t((voho~)x)

is continuous.

But this follows from the

fact that (voho~)x = vohox*~ and the functional A + t(vohoA) is in M(S)*. Similarly voho~ £ LWUC(S) and we are done. We now prove the main result of this section.

85

Let 5 be a C-distinguished topological semigPoup admitting a non-aePO absolutely continuous measUPe (i.e. Ma(S) is non-zero). Then if 5 is not Nlatively neo-compaot, we have that the quotient space WUC(S)fwAP(S) contains a lineaP isometPic copy of 1~ and so is nonsepaPable. 3.3 THEOREM.

~

As noted in item 1.5, we trivially have WAP(S)

Proof.

WUC(S).

Since Ma(S) is solid (by Theorem 2.1.7), we can choose a positive measure n

Ma(S) such that

£

X ~

v*x and

X ~ x*~

measures, v and

~.

and K : .. supp(n) is compact.

By Corollary

such that if v := n*u and ~ := v*n then the maps

£ 5

of 5 into M(S) are weakly continuous.

We keep these

fixed for the remainder of our proof and note that both

C := supp(v) and D := 0

=1

llnll

2.1.13 we can find u,v

0

supp(~)

are compact, (in fact C .

0

= Ku

and D = vK), 0

and II v II ... II ~ II ... 1. Let C :• C0 u{l} and D :• D0 U{l}, where 1 is the identity of 5 (if there is one) or an adjoined isolated identity of S. Let sequences A :• {x1 ,x2 , ••. } and B := {y1 ,y 2 , ••• } be chosen as in Lemma 3.1 with respect to the compact sets C and D. We can choose infinite subsequences~

:•

{~ ·~

1 ~

, ••• } of A and Bk :- {yk ,yk , ••• } of B such that 2

1

2

~

(a)

and

u Bk k=l

then

An n Am

C

B;

(B)

if n I< m

(y)

Lemma 3.1 remains valid with k ,k ,k. and k. in place of n

n,m,i and j, respectively (i.e.

m

J

1

when~

and Bk replace A

and B, respectively). Let ~

~

:•

~

and

We define the functions fk on 5 by

Let {ck} be any element in 1~.

86

Fk ·= •

U

U

j=l i>j

C~.Yk.D. 1

J

.

We now show that

I

ckfk is in WUC(S).

From (B) and (y) we note that

k=l all the

~·s

and Fk's are pairwise disjoint and so can define the functional

h e: M(S)* by

.

Now a simple exercise on our definitions shows that

..

and so

.

I ckfk e: WUC(S) by Lemma 3.2. k=l

It remains to show the (clearly linear) map

.

..

is an isometry of 1 demonstrating that

where f :=

.. I

~fk

into WUC(S)/WAP(S)"

and g e: WAP(S).

To achieve this we start by

Suppose on the contrary there exists

k•l a g in WAP(S) and e: > 0 such that

We can find a positive integer k' such that ~·

may assume that (2)

llf+gll S

<

Ck'

ll
-

E

<

lck' I·

We

is non-negative and hence obtain -Eo

From the definition of f :=

.. I

ckfk

we have that

k=l

87

if

i~j

if

i > j

(3)

From (2) and (3) we obtain

{~ ~ ~

(4)

1

> J

implies

g(~! Yk!) ~ -ck' + lck,+g(~! Yk!>l < J

1

implies

g(~! Yk!) ~ ck' - 1-ck,+g(~! yk!>l > J

1

£

J

1

1

£.

J

From (4) and Proposition 1.2. we have that g 4 WAP(S), which contradicts our original choice of gin WAP(S). Hence (1) holds. Noting that ao

and recalling (1), we have that ao

II k..I l

ao

~fk + WAP(S) II WUC(S) I

g

E:

WAP(S)}

WAP(S)

ao

I ~fk + WAP(S) is a linear isometry of k•l into WUC(S)fwAP(S) and our proof is complete.

Consequently the mapping {ck} Lao

+

LetS be a non reZativeZy neo-compact stip with Ma(S) nonaero. (In partiouZar S may be any ZocaZZy compact group which is not compact.) Then the quotient space UC(S)fwAP(S) contains an isometric Zinear co'PiJ of tao. 3.4 COROLLARY.

Proof. We have UC(S) • WUC(S), by Theorem l.S(i), and so our result follows from Theorem 3.3. Our next result and Theorem 3.7 are versions of Theorem 3.3 with proofs essentially contained in that of Theorem 3.3.

88

Let S be a C-distinguished topological semigroup, p a positive measure in Ma(S) with compact support, IIPII = 1 and suppose that S is not relatively neo-compact. Then WUC(S)op\ WAP(S) is non-separable (in C(S)). 3.5 PROPOSITION.

Proof.

F :=

Let

A:= F n

norm-cl(WUC(S)op),

WAP(S) and E

:=

supp(p).

In

the proof of Theorem 3.3 take D := (Eu{l})(D 0 u{l}) and let the functions fk be as constructed there.

is an isometry of 1~ into

Then the (clearly linear) map

F;A

and hence

F \ A is

non-separable.

This implies

our result. For the purpose of our next lemma, we recall that if A is any set of functionals then A+ := {a E A : a~ 0} and A- := {a E A : a~ 0}. L

For any n E M(S), let n := {hE M(S)* h(v*n) = 0 for all Then if p and n are any positive measures in M(S) with p << n,

3.6 LEMMA.

v E Ma(S)}. we have that L

(n )

+

S

L

(p )

+

• ~

Proof.

LethE (n)

+

We may assume that v = v 1-v 2 , where Then for i = 1 or 2, v 1 and v 2 are positive measures in Ma(S). h(vi*n)

and so h(vi*x)

=

J h(vi*x)dn(x)

=0

continuous function. h(v.*p) 1

=f

L

and v E Ma(S).

=o

(x E supp(n)) since x ~ h(vi*x) is a non-negative Since p << n we have supp(p)

h(v.*x)dp(x) 1

= o.

5

supp(~)

and so

So we conclude that

+

and h E (p ) •

89

Let s be a c-distinguished topoZogiaaZ semigroup, n a positive in Ma(S) with llnll • 1, and suppose that

3.7 THEOREM. meas~e

WUC(S) S norm-cl(WAP(S) + n

J.

+

X)

Then s is 'l'eZativeZy neo-aompaat.

fo'l' some no:rm sepa'l'abZe X S Ma(S)*. Proof.

Let

that ~ «

~ £

n and

WUC(S)

+

Ma(S) be a positive measure with compact support and such II~

II '"'

1.

By Lemma 3.6,

S norm-cl(WAP(S)

+

(~

+

.L

)

+

+ X1 )

-

... +

for some norm separable x1 c M (S)*. For each h £ (~ ) and v £ M(S), we - a have v*~ £ M (S), so he~ 2 (v) := h(v*~ 2 ) ~ h((v*~)*~) s 0. Consequently • a zero and , tak"1ng p := ~ 2 , we get ( ~ .L)+ e ~ 2 1s WUC(S)

+

e p Snorm-cl(WAP(S)

+

e p +

for some norm separable X' c M (S)*. - a separable X'' c M (S)* such that -

WUC(S) Hence WUC(S)

x1

e p) S norm-cl(WAP(S) + X')

Similarly, we can find a norm

a

e p S norm-cl(WAP(S) +X''), 0

p \ WAP(S) is separable (in the norm of Ma(S)*).

Now the norm of Ma (S)* is in fact

II II P(M

(S)); so for all f and g in

a

WUC(S). noting that x*p llfop 2-gop 2 11

s

£

P(Ma (S)) whenever

:= sup< lf<x*p 2> <

- g(x*p 2>1

sup{ Ifep (v) - gep (v) I

:

X £

s.

we th"JS have

X £ S}

v

£

P(Ma (S))} =

II fep-gepll P(M

(S)) •

a

Therefore WUC(S)op 2 \ WAP(S) is separable with respect to the norm of C(S), By Proposition 3.5, it follows that S is relatively neo-compact. In the case of groups from Theorems 3.7 and 1.5, we have the following improvement of a result of E.E. Granirer [so].

90

3.8 COROLLARY. Let G be any locall.y compact topological gl'Oup and 1 a E L (G). a> 0 and llall = 1. Let aL := {f E L00 (G) : (f 1 L1 (G)*a) -

If UC(G) ~ norm-cl(WAP(G) +a G is compact.

L

O}.

+X) fo~ some norm sepa~le X~ L (G)

then

00

1

For ease of reference we mention the following consequence of Thoerem 3.3

Let s be a c-distinguished topological semigl'Oup with M (S) ~ ~ a non-ael'O and such that C D and DC aN compact fo~ all compact subsets c and D of S. Then S is locally compact and the following items aN equivalen; 3. 9 COROLLARY.

(i)

s is compact;

(ii) C(S) = WAP(S); (iii) WUC(S) 3.10 Notes.

= WAP(S). Let G be a locally compact topological group.

proved that C(G)

= WAP(G)

[49] improved this to:

if and only if G is compact.

UC(G)

= WAP(G)

Burckel [13]

Then E.E. Granirer

if and only if G is compact.

Generalizing these results Dzinotyiweyi [35] proved Corollary 3.9.

For an

amenable group (i.e. a group G with UC(G) admitting an invariant mean). Corollaries 3.4 and 3.8 were proved by Granirer in [49] and [so] • respective: Also Ching Chou

[18] proved Corollary 3.4 for groups satisfying a certain

condition he calls property (E).

The generalization of these results given

in Theorems 3.3 and 3.7 is taken from Dzinotyiweyi [37]. It is natural to ask what happens for the situation where M (S) is zero. a For instance. a close examination of Corollary 3.9 leads us to the following conjecture raised in [36]. CONJECTURE A. -1

Let S be a locally compact semi-topological semigroup such

that C D and DC

-1

are compact for all compact subsets C and D of S.

Then

the following items are equivalent: (a) S is pseudocompact 1 (b) C(S)

= WAP(S).

(c) WUC(S)

= WAP(S).

Evidently

(b) implies

(c) and below we show that (a) implies (b).

the conjecture will be solved if one can show that (c) implies (a).

Thus

However. 91

(if this turns out to be difficult), it would also be interesting if one could show that (b) implies (a). PROPOSITION. Proof.

We have that

(a)

impZies (b).

It is a simple exercise to show that every locally compact pseudo-

compact space is countably compact.

(See e.g. Gillman and Jerison [44].)

This together with Lemma 4.2 implies our result. We also have

If in addition to the hypothesis of oUl' conjecturoe S is a-compact~ then C(S) = WAP(S) if and only if s is compact. (In fact the actual result proved in [36] is more general.)

THEOREM [36].

Thus our conjecture is known for the case where S is a topological semigroup admitting a non-zero absolutely continuous measure (see Corollary 3.9) or S is a-compact.

In general item (b) does not imply S is compact as we

note in the following example. EXAMPLE [35].

Recalling the usual well-ordering of the set of all ordinals

and taking w1 to be the first uncountable ordinal, let S := {a : a is an ordinal and a< w1 } with the usual interval topology and maximum operation.

Now one can note

that

• a 1oca 11 y compact topo 1og1ca · 1 sem1group · · h (1• > S 1s w1t

c-1o

and Dc- 1

compact for all compact subsets C and D of S; (ii) S is pseudocompact but not compact (see e.g. [44]); (iii) Ma(S). is zero; (iv) C(S) • WAP(S),

(by (ii) and the above Proposition).

We have already noted that a stip S behaved as if Ma(S) were non-zero. So the following conjecture seems reasonable. CONJECTURE B [37].

If Sis a stip that is not relatively neo-compact, the

quotient space UC(S)fwAP(S) contains an isometric linear copy of 1~.

92

Even if it turns out that every stip S is such that M (S) is non-zero, it a would be interesting to solve this conjecture without the use of M (S). We a suspect that the notion of functions with separable orbits may be useful in finding a function in UC(S)\WAP(S). 4. THE STONE-~ECH COMPACTIFICATION OF A SEMIGROUP

In this section semigroup.

~e

take s to be any compLeteLy reguL2r semitopoLogicaL

We now define the Stone-~ech compactification BS of S.

For each

f E C(S), let Df :={A E C: IAI ~ llfll S} and note that Df is a compact disc. Hence, by Tychonoff's theorem, E := x{Df : f E C(S)} is compact. In a natural manner we embed S into E via the mapping 6 : S + E given by 6(x)(f) := f(x) Then BS := cl(6(S)).

for all

xES

and

f E C(S).

For our convenience we shall write x in place of 6(x)

(and hence S in place of 6(S)), for all x inS. Suppose that S is locally compact.

Then C(S)* is the space of bounded

finitely additive regular Borel measures on S.

It is interesting to study

C(S)* as a Banach algebra with an Arens or convolution product.

This idea

is suggested in Hewitt and Ross [59, page 275] and has been studied to some extend by Butcher [14], Olubummo ( [75J and [76]) and Pym and Vasudema <[84] and [85]).

Now, since C(S) can be identified with C(BS), the Riesz

representation theorem enables us to identify C(S)* with M(BS).

So one may

obtain some information about C(S)* by studying BS as a semigroup. First we introduce an "Arens operation" on BS:

For each a E BS and

f E C(S), we define the function aof on S by aof(x) := a(xf) for all x in S.

For all a,B E BS we define the Arens operation, o, by aoB(f) := a(Bof)

for all

f E C(S).

It is clear that this operation is well defined if and only if Bof E C(S) for all BE C(S) and f E C(S). Let LMC(S) := {f E C(S) : aof E C(S) for all a E SS}.

Then we have the

following simple exercise for the reader.

93

4.1 PROPOSITION.

The foZZowing items are equivaZent:

(i) (SS,o) is a 1,eft semitopoZogicaZ semigPOup; (ii) LMC(S)

= C(S).

Next we are interested in the topological structure of semigroups S for which (SS,o) is a left semitopological semigroup.

Partly towards this end

we include the following lemma.

4.2 LEMMA. Proof.

If sis countabZy compact then C(S)

Let {xn} and {ym} be sequences in S.

= WAP(S).

If S is countably compact,

we can find cluster points x and y in S of {xn} and {ym}, respectively. for each f

E

So

C(S) we clearly have f(xy) in the closure of each of the sets

{f(xnym) n < m} and {f(xnym) : n > m}. have f E WAP(S) and our result follows.

Recalling Proposition 1.2 we thus

4.3 THEOREM. Let s be (topoZogicaZZy) no~z~ ZocaZZy compact~ right canceZZative and such that c-1n is reZativeZy compact for aZZ compact subsets C and D of s. Then the foZZo~ing items are equivaZent: (i) (SS,o) is a Zeft semitopoZogicaZ semigroup; (ii) Proof.

s is discrete or countabZy compact. From our definitions and lemma 4.2, it is trivial to note that

item (ii) implies that LMC(S)

= C(S)

and hence item (i) follows by

Proposition 4.1. Now suppose that (i) holds with S neither discrete nor countably compact. We can find a relatively compact infinite set

{sn : n

E

m} in S and a

sequence {Un } of subsets of S such that Un is a neighbourhood of s n and sm ~ Un for m n. Let C := cl({sn : n E m}) and note that C is compact. We can choose a sequence {tn } in S without a cluster point and such that

+

for all

n

E

m.

(1)

Next we consider the closure X of the set X : = {smtn : m, n (sm(a)tn(a)) be a set in X converging to x

E

S.

E

m}.

Let

Let D be a compact neigh-

bourhood of x and choose a 0 such that a~ a 0 implies that sm(a)tn(a) ED. 94

-1 Since c- 10 is relatively compact and {tn} Thus tn(a) E C D for a ~ a 0 • has no cluster point, we must have {t ( ) : a > a } finite. Hence x = ctn n a - o for some n E 1N and c E C. The representation x = ctn is unique, for if ctn

c 1 tm for some c 1

E

C and m

E

1N , then n

= m by

(1) and so c • c 1 by

right cancellation. It is now clear that if we define f on X by setting f(smtn) := Jl

l_o

f(ctn) :=

0

n

if

m

if

m> n

<

otherwise

then f is continuous.

(for c

€

C),

By the Tietze extension theorem, f extends to a

continuous function, g say, in C(S).

Hence, by Theorem 4.4, we have that

g ~ LMC(S) and so (i) does not hold, by Proposition 4.1.

By this contrad-

iction our proof is complete. Since LMC(S) plays a significant role in the study of SS as a semigroup, the following characterizations of LMC(S) are of interest in this section. First we recall that a (Hausdorff) topological space X is a k -space if 0

every complex-val•1ed function on X that is continuous on compact subsets of

X is continuous on X.

(Note that a k 0 -space is not necessarily a k-space

- see e.g. Exercise 4.13.) I f cj>

C(S)* is such that +(f .g)

E

= cj>(f)+(g)

LetS be a k 0 -space and f 4.4 THEOREM. Ol" equivaZ.ent: (i) f

E

E

we say cj> is muUiplicative.

C(S).

Then the

foZ.Z.o~ing

items

LMC(S).

Fol" aZ.Z. sequences {xn} and {yn} in S ~th {xn n E 1N} l"eZ.ativeZ.y compact~ ~e have the cZ.osul"es of the sets {f(xnym) : n < m} and {f(xnm y ) n > m} not disjoint. (ii)

(iii) { f X

(iv) f

E

x

E

K} is

~eakZ.y

compact fol"

eve~

compact K c s.

LWUC(S).

95

: n Em}) and let n P be the set of all multiplicative means on C(S) endowed with the weak*Proof.

To show that (i) implies (ii), let K := cl({x

topology.

Thus both K and P are compact.

F : K x P

~

For f

E

LMC(S), we have the map

C, given by

F(x,n) := n(xf)

for all

(x,n)

E

K x P,

separately continuous. Assuming the notation of (ii), let x' be a cluster point of {xn} inK and v a cluster point of

{ym} in

P.

Consequently every neighbourhood of v( ,f) x meets infinitely many columns and infinitely many rows of the double So the closures of the sets {x f(ym) : n < m} and n {x f(ym) : n > m} in C are not disjoint. Thus (i) implies (ii). n

The equivalence of (ii) and (iii) follows from Grothendieck's Theorem (-see Appendix B.7). That (iii) and (iv) are equivalent is left as a simple exercise for the reader. Now from Theorem 4.4 and the proof of Theorem 4.3 we note that if a semigroup S satisfies the hypothesis of Theorem 4.3, then C(S) only if S is either discrete or countably compact.

LWUC(S) if and

We now show that an

easy adjustment of the proof of Theorem 4.3 gives a more general result of independent interest.

4.5 THEOREM. Let s be no~az~ ZocaZZy compact and right JanceZZative. Suppose S is neither countabZy compact nor discrete and c- 1n is compact for aZZ compact subsets C and D of s. Then~ for some cZosed subset Xof s~ ~e have that (C(S)\LWUC(S)) IX contains a Zinear isometric copy oft~ and so the quotient space C(S)fLWUC(S) is non-separabLe.

Proof.

We urge the reader to first study the proof of Theorem 4.3 as we

will omit closely related details in this proof. Let thesequences {s } and {t } and the set C be as constructed in the n

proof of Theorem 4.3.

n

Choose infinite subsequences Tk :s {tk ,tk , ••• } of 1 2 T := {t 1 ,t 2 , •••• } such that 96

.

u Tk '"' T, and k=l Tk n Tk' =; Let

~

:= {smtk n

k I k'.

if and only if

: m,n E 1N}, X :• {smn t : m,n

E

:tl}

and note that our

construction of the Tk's and T imply (a) ~ ~

(b)

..

(c)

c

C and

E

n

E

lN} •

n ~· ~;if and only if k I k', ~

u

k=l

= X.

(To verify these items, see proof of Theorem 4.3 for relevant techniques.)

Next we define the functions fk :

fk(sm.tk ) m

·-r . -1

fk(ctk ) := -1 n

if

if

m< n

if

m> n

~ E

lR by

-

C E

C \ {sm

m

E lN} •

Then (as similarly shown in the proof of Theorem 4.3) fk is continuous, for all k

E

':tl.

Corresponding to each element {~,} in t."", let F(~,) be the

function defined on X by for some I. By items (b) and (c) we thus have

F(~,)

lN.

E

well defined as a function.

To see that F(~,) is continuous, suppose (catn(a)) is a net in converging to some point

c~

1

, for some c a • c

by the definition of T, we have n(a) = k (c t ) c ana -

X•

E

C and t na ( )' tk 1

E

X T.

Then,

eventually and so, eventually

Consequently, eventually

97

Thus F(dk,) is continuous. Now noting that if

m < n

if

m > n

(*)

Theorem 4.4 and Tietze's Extension Theorem imply the existence of a function F(dk,) £ C(S)\LWUC(S) such that F(dk,>lx=F(dk,)

and

IIF(dk,)lls=

IIF(dk,)llx=ll{dk,liiQ).

Thus the (clearly) linear map {dk,} ~ F(dk,) li is isometric.

of tQ) into (C(S)\LWUC(S))Ii

Since tQ) is non-separable it follows that C(S)\LWUC(S) and hence C(S)fLWUC(S) is non-separable. Theorems 4.3 and 4.4 are taken from Baker and Butcher [7].

The argument

used in showing that (i) implies (ii) in the proof of Theorem 4.4 is inspired by the proof of [106, Theorem 1]. Also in the paper [7], Baker and Butcher showed that SS is a left semitopological semigroup under a certain operation extending that of S while C(S) # LMC(S), for many locally compact semitopological semigroups S.

Such

arbitrary products unfortunately do not seem to reflect much on the structure of SS or LMC(S). of interest.

On the other hand the Arens type of products seem to be

In what follows, we make use of Arens products defined wi·th

respect to various subspaces of C(S). 4.6 NOTATION.

For the remainder of this section let AS£ {LMC(S), LWUC(S),

WUC(S), C(S)} and A(S) be the compact space constructed with respect to AS as we constructed SS with respect to C(S).

(Thus A(S) is the spectrum of

the algebra AS.) For all f £As and a£ A(S), let aof be the function on S 4.7 EXERCISES. given by aof(x) := a(xf) for all x £ S. Then if As £ LMC(S) we have (i) aof £ As

98

(ii) (A(S),o) is a left semitopological semigroup with the Arens operation 'o' given by aoS(f) := a(Sof), for all a,S 4.8 PROPOSITION.

The

foZZo~ing

E

A(S).

items are equivalent:

(i) A(S) is a semitopoZogicaZ semigPoup

~th

subsemigroup

s;

(ii) AS= WAP(S). Proof.

That (i) implies (ii) follows from our definition of A(S) and

Lemma 4.2.

Now suppose (ii) holds and consider the semigroup (A(S),o)

defined in Exercise 4.7(ii).

To verify item (i) it is sufficient to show

that (A(S),o) is a right semitopological semigroup.

To this end, let f

be fixed and suppose (a) is a net converging to a in A(S). y

E

AS

We have

Hence (ay of) S} compact and (a of) c Cf. y has a weak cluster point g (say) in As. In particular for all x in S we Cf := weak-closure of { f : x X

E

have that a y of(x) := a y ( X f)

~

a( X f)

= aof(x)

and g(x) is a cluster point of (a y of(x)). Hence g = aof. It follows that a~of ~ aof in the a(AS,A~)-topology. So for all S E A(S), we have

Thus (A(S),o) is a right semitopological semigroup.

Let S be Locally compact and such that c-1D and DC-l ape If S is eitheP a-compact compact foP aU compact subsets C and D of S. and As = C(S) oP S is a topological semigroup suppopting a non-aero absolutely continuous measUPe~ ~e have the foZZo~ing items equivalent: 4.9 THEOREM.

(i) A(S) is a semitopoZogicaZ semigroup

~th

subsemigroup S;

(ii) S is compact.

Proof.

This is an immediate consequence of Corollary 3.9, the Theorem

mentioned in item 3.10 and Proposition 4.8. The advantage of the compactification A(S) of S is that already, if AS E {LMC(S), LWUC(S), WUC(S)}, A(S) is a left semitopological semigroup. Our preceding Theorem now says that, for a large class of non-compact S, 99

A(S) is not a semitopological semigroup.

A natural question to ask is:

What is the largest semitopological semigroup BA(S) contained in (A(S),o) and such that S is a subsemigroup of BA(S)? first introduce the following notation.

To formalize this question we

Let

BA (S) : = {S E A(S) : the map a + Boa of A(S) into A(S) is continuous}. BA(S) is called the set of bicontinuous points in A(S) in [88] (where S is a group).

4.10 EXERCISE.

We always

haveS~

BA(S).

The following conjecture seems reasonable. Let S be locally compact and such that c- 1D and DC-l are 4.11 CONJECTURE. compact for all compact subsets C and D of S. Then if S is either a-compact or such that S is a topological semigroup admitting a non-zero absolutely continuous measure, we have BA(S)

= S.

For the case of a discrete topological abelian group, the above conjecture is due to Baker and Milnes

[a].

4.12 THEOREM (Ruppert [88]). foz:Lowing cases:

We quote the following cases now known.

We have that BA (S)

S for each of the

s is an abeZian ZocaZZy compact topoZogicaZ group; s is a connected ZocaZZy compact topoZogicaZ group; s is a discrete topoZogicaZ group with the set of a~Z eZements in s

(i)

(ii) (iii)

of order tw countabZe. We conclude this section by giving the reader an exercise promised before.

4.13 EXERCISE.. In general every k-space is a k 0 -space. We required S to be a k -space in Theorem 4.4. We now mention the following semitopological 0 semigroup S which is a k -space but not a k-space. 0 Let

T

0\P

with 0 open in the usual topology of 1R and P a countable subset of 1R.

Let 1R

be the topology on the real numbers with generic open sets of the type

T

denote the resultant space.

Then with the additive operation S := 1RT

is a semitopological semigroup that is a k 0 -space but not a k-space, e.g. [9] for details.)

100

(See

5. REGULARITY OF MULTIPLICATION IN SEMIGROUP ALGEBRAS Let A be a Banach algebra with first and second dual spaces A* and A**• respectively. hf in A* by voh(~)

For f

:=

€

h(v.~)

A**• h

€

A* and v

hov(~)

and

:=

€

A we define voh, hov, hf and

h(~.v)

(~ € A) (~ €

A).

We then define the Arens products ( [1] and [2]) e and e' on A** by

for all

f,~ €

A** and h

€

A*.

If the two Arens products coincide (i.e. we say multiplication in A is regular.

irregular if it is not regular.

f•~

=

f•'~

for all

f.~

in A**)

Multiplication in A is said to be

The products were first introduced by Arens

and have been used in various publications.

It is unusual to find a

property of Banach algebras that is invariant under passage both to subalgebras and to quotient algebras. of regularity of multiplication.

One such invariant property is that In this section we are interested in

finding when we have this property for algebras like Ma(S).

With respect to either product • ore', A**

is a Banach algebra into

which A is embedded isometrically by the canonical homomorphism v (v

€

~

n(v)

A) where n(v) (h) : = h(v)

(h

€

A*).

It is immediate from the definitions that either Arens product on A** is Any pair {n(v )}, {n(~ )} of separately continuous for a(A**• A*). n m bounded sequences in n(A) will have cluster points f,~ (respectively) in A** for a(A**,A*). h

€

If multiplication in A is regular, it follows that if

A* is such that both repeated limits of the double sequence {h(v

exist, then these two limits are equal (being in fact

n

·~

)} m

f•~(h)).

Recalling Proposition 1.2, we summarise the above remarks (for measure algebras) in the following lemma. 101

5.1 LEMMA. Let s be a c-distinguished topoLogicaL semig~up and A a cLosed subaLgebra of M(S). Then the foLLowing items are equivaLent: (i)

A has reguLar muLtipLication;

(ii) if {vn }, {~m} are any bounded sequences in A and h £ A* such that b := lim lim h(vn *~m) and c :~ lim lim h(vn *~m) e~st, then b = c; m n n m (iii) evePy h

£

semigroup (P(A),

A* is a weakLy aLmost periodic fUnction on the topoLogicaL

II II).

(Recall the definition of P(A) from 2.1.)

For any A S M(S) we recall that dA denotes the density function as defined in item 1.3.2.

be a c-distinguished topoLogicaL semig~up with identity eLement 1 and A a convoLution measure aLgebra with foundation equaL to s. Let W be a subset of s such that 1 is not isoZated in W and WS dA(S). Then there ~st sequences {Cn}• {Dm} of non-A-negLigibLe compact subsets of W such that, for aZZ n,m,i,j £ 1N with n < m and i > j we have 5.2 LEMMA.

Let

S

Cn Dn n C.D. = 1 J

Proof.

+•

Suppose, by the induction hypothesis, for some positive integer p

we have finite sequences {c1 ,c 2 , ••• ,Cp}• {D1 ,n 2 , ••• ,Dp} of non-A-negligible compact subsets of W such that if p-·1 p p X := u cn• y :• u Dn• L := u u cn Dm p p p n=l n=l n<m_3) n""l and p-1

:= u p m=l

u

R

p~n>m

cn Dm

then (a) 1 • X

p U Yp u Lp u Rp

(B) xp n

YP

(y 1 ) LP n YP

~

(y 2) RP n xP

+ +

(15) Lp n Rp •

102

+

To prove the inductive step, we now choose Cp+l and Dp+l and verify items (a) to (~) with p+l in place of p. (Note that all sets mentioned in items (a) to (6) are compact.)

We can choose a non-A-negligible compact set Cp+l c W such that (1)

1 ~ cp+l u cp+l YP

(2)

cp+l n (YP u RP) =

(J)

cp+lyp n (LP u XP u cp+l) •

by (a)

+

by (a)

+

by (B), (y 1) and since 1 ~ YP.

Next we can choose a non-A-negligible compact set D 1 c W such that p+

(4)

1 1 D

by (a)

(5)

Dp+l n (XpDp+ l u Lp u Xp+ 1)

=•

..

p+l uXD p p+l

by (a) and since 1 • Cp+ 1 •

We are now in a position to establish items (a) to (6) with p+l in place of p. Noting that Lp+l ~ Lp u XpDp+l and Rp+l = Rp u Cp+ 1Yp , it follows from (a), (1) and (4) that 1 ~ Xp+l u Yp+l u Lp+l u Rp+l.

From (8), (2) and (5) we get

From (y 1 ), (5) and (6) we get (Lp U Xp Dp+ 1) n Yp+l

From

(~),

(J) and (6) we have

p+l n Rp+l

L

=

(L

p u XpDp+ 1 ) n (Rp u Cp +ly p )

.. (Lp n Rp) u (Lp n Cp+lYp) u (XpDp+l n (Rp

U

Cp+l Yp))

- cf>. 103

This completes our verification of the inductive step.

By repeating the

argument countably many times we note, in particular, that item (6) leads to the conclusion of our lemma.

Let S denote a c-distinguished topoZogicaZ semig~up with an identity eZement 1 and A a convoLution measupe aZgebra with foundation equaZ to s. Then 5.3 THEOREM.

(i)

If

1

is not isoZated in s we have muZtipZication in A iX"l'eguZar.

If the foundation of Ma(S) coincides with S and the sets x -l{y} and 1 {y}x- are finite (x,y e S), then A has reguZar muZtipZication if and onZy (ii)

if

S

is finite.

Proof.

To prove (i) suppose that 1 is not isolated and choose sequences

{Cn }, {Dn } as in Lemma 5.2 (with W := S). positive measures vn .~ n in

II vnll = vn(Cn) = 1

A such and

Since

A is

solid, we can find

that

n~nll = ~ n (Dn ) = 1

(n e 1N).

00

Since L := u u CD nm is a-compact, we can define h n""l n<m h(n) := n(H) (n e A).

E

A* by

We then get h(vn *~m) := vn *~m(L)

=

if

n < m

if

n

>

m.

Recalling Lemma 5.1, our proof for item (i) is complete. To prove item (ii) we first remark that if S is finite then trivially multiplication in

A is

regular.

case where S is discrete. Lemma 3.1(a)· (with C

=D=

For the converse we first consider the

Now if S is discrete and infinite, recalling {1}), an argument similar to that used in the

proof of item (i) (which we omit) easily shows that multiplication.

Thus S must be finite if

A has

A has

irregular

regular multiplication.

Finally we consider the case where S is not discrete.

In view of item (i)

our proof will be complete if we can show that 1 is not isolated.

Since S

is the foundation of Ma(S), if 1 is isolated than I e Ma(S) and Sis discrete -see e.g. Exercise 2.3.10(c)(ii). By this conflict if Sis not discrete, 1 104

is not isolated and our result follows.

Let A be a convolution measuPe algebPa whose foundation (semigPOup) A is a topological subsemigPOup of a topological gpoup G. the following items aPe equivalent: 5 • 4 THEOREM.

(i) Multiplication in

Then

A is PegulaP;

(ii) S is finite. Proof.

That (ii) implies (i) is trivial.

Suppose (i) holds.

assume that S is closed in G for otherwise we may take

S in

We may

place of S and

note that S is a closed subsemigroup which we may also take to be the foundation of A.

If S is discrete, that (i) implies (ii) can be verified as

done in Theorem 5.3(ii). Suppose S is not discrete.

If S is not compact then by arguing in a

manner simpler to the proof of Lemma 3.l(a) we can find sequences {C }, {D } n m of non-A-negligible compact subsets of S such that n < m and i > j imply that C D n C. D. = 4» n m 1. J

(n,m,i,j

£

1N).

Recalling our proof of Theorem 5.3(i), this similarly leads us to a contradiction.

Next we suppose that S is compact.

Then S is a compact group,

by Theorem 1.2.3, and Theorem 5.3(ii) gives us the result. 5.5 ~·

(i) Civin and Yood [19] proved that if G is an infinite abelian locally compact topological group, multiplication in L1 (G) is irregular.

Then Young [105] improved the result to include the case where G is not necessarily abelian.

Pym [81] studied the regularity of multiplication

for certain convolution measure algebras supported on semigroups and so did Young [106].

Motivated by these results Dzinotyiweyi [35] proved the

preceding two Theorems.

In particular Theorem 5.4 is a generalization of

Theorem 5 of Pym [81] • (ii) At this stage it should be clear to the reader that: of continuity of multiplication in

as.

the phenomenon

the weakly almost periodicity of

functions on S and the regularity of multiplication of convolution measure algebras "living" on S, are somewhat related. Indeed if Sd denotes the semigroup S with discrete topology, then the reader may easily extract a proof for the following Theorem of Young [106] from Sections 3,4 and 5.

105

The

THEOREM.

foll~ng aPe

equivalent for any c-distinguished topological

semigroup S: (a) M(S) has regular multiplication; (b) t 1 (s)

has regular multiplication;

(c) BSd admits the structure of a semitopotogical semigroup

~ith

S as a

subsemigroup; there is no pair of sequences {xn }, {ym} in {x y y : n > m} are disjoint; nm : n < m} and {xnm (d)

(e) m(S)

S

such that the sets

= WAP(Sd).

(iii) A major difference between Lemma 3.l(a) and Lemma 5.2 is that;

in

the former sequences {x }, {y } are chosen in a "scattered manner" where as n m in Lemma 5.2 the sequences {C }, {D } are chosen within a given "vicinity" n m of the identity element of S. Now letS := [0,1] with maximum operation. Then S = 1-1 {1} and Sd is relatively neo-compact.

However, from Lemma 5.2 and the above Theorem, we

have that BSd is not a semitopological semigroup with subsemigroup S.

Thus

the Stone-Cech compactification of a locally compact relatively neo-compact topological semigroup is not necessarily a "nice" semitopological semigroup. (iv) THE RADICAL OF Ma(S)**·

Throughout this item S denotes a

foundation semigroup with identity element 1 (unless otherwise stated) and Ra (S) the radical of the Banach algebra Ma (S)** with Arens multiplicaton •'· For convenience we assume all spaces to be real. We are interested in showing that, for many cases, the space Ra(S) is very large.

Towards this

end we first prove the following result, also of independent interest.

Let S be non-discrete and right cancellative. Then the quotient spaces Ma(S)*fc(S) and Ma(S)*fLWUC(S) contain isometric linear copies of t"". PROPOSITION.

Proof.

Let W be a compact neighbourhood of 1 and corresponding to each

function g in C(S) let G be the function in G(x,y) := g(xy)

106

for all

x,y

£

W.

C(W~W)

given by

Then a simple compactness argument shows that the set {G(x,.) : x

£

W}

is

relatively (norm and hence) weakly compact in C(W). We can find a sequence {Vk} of disjoint open neighbourhoods contained in w. Choose vk £ Vk n s1 and recall that Vkv;1 is a neighbourhood of 1. So there is a sequence {Uk} of open neighbourhoods of 1 such that U~ ~ Vkv~ 1 • for all k £ B. By Lemma 5.2 we can choose sequences {~ }, {Dk} of nonn

n

Ma(S)-negligible compact subsets of Uk such that, for all n,m,i,j

£

~.

we

have ~

n

Dk n m

~.

Dk.

1

J

=+

whenever n < m and i > j.

By right cancellation we have Ck

n

:a

Dk vk n ck. Dk. vk m 1 J ~a(S)

and choose sequences of points {ck }, {ek } such that n

ck n

£

(1)

+whenever n < m and i > j.

da(~)

and

n

ek n

£

da(Dk

n

(2)

vk).

n

...

...

u u ck. Dk. vk, and define the Let Ek := u Dk. vk and Fk := u i>j i .. l i<j ~.1 1 j-1 J J

function

~

Noting that

by ~ £

~

:= X~ - XFk

.

Ma(S)* we now show that (in the norm of Ma(S)*)

II~+ gil

> 1

If not, for some g

£

for all g

£

(3)

C(S).

C(S) we can find

£

> 0 such that

II~ + gil ~ 1-e

In particular for

va,~B £

P(Ma(S)) with supp(va) C

~n

and

supp(~B) C

Dkm vk,

we have

and so (recalling our definition of

~)

107

Jif n < m then 11 + g(va.*lJs) I ~ 1-£

lif n > m then 1-1 + g(va.*JJs>l ~ 1-£. Letting (va.) shrink to ck

and (JJB) to ~ we thus get (by continuity of g)

n

{

m

if n < m then 11 + g(ck ek ) I ~ 1-£

n m if n > m then 1-1 + g(~ ek >I < 1-£. n m I t follows that < -£

if n

>

if n > m

<

m

{ £

W} is not relatively weakly compact, by Grothendieck's Theorem (-see Appendix B.7). This contradicts the observation at the beginning of our proof. By this conflict item (3) holds. and so {G(x,.) : x

E

Since the Vk' s are pairwise disjoint and

~

n

oo

n,m,k

E ~

we have that {tk}

E

~

tk~

Dk

vk c Vk for all

m

+ C(S) defines a linear mapping of

ksl R-00

into Ma(S)*fc(S)·

Noting that

11~11

= 1,

item (3) implies that

00

00

and so the mapping {tk}

~

E

tk~

+ C(S) is isometric.

k=l 00

Similarly the mapping {tk}

~

E

~~ +

LWUC(S) of R.

00

k=l is linear and isometric.

THEOREM. Let S be a nondiscPete and Pight canceZLative foundation semigPOup with an identity element. Then thepe exists a subspace P of Ma (S)* such that P* is a lineaP isometPic copy of (R.00 )* and the PestPiction of the Padical of Ma (S)** to P is P*. In paPticulaP the Padical of Ma (S)** is nonsepaPable.

108

Let A :

Proof.

3

{~ £

Ma (S)** :

~(f)

• 0 for all f in LWUC(S)}.

~ £

Ma(S)**• v £ Ma(S) and h £ Ma(S)*, we have that voh left handed version of) Lemma 4.3.2; consequently

h~(v)

:=

~(voh)

= 0

£

For all LWUC(S), by (the

and so (~ £

A>.

Thus A is a right ideal of Ma (S)** such that Ao'Ma (S)** is zero.

Hence

A c R (S), (see e.g. Rickart, "General theory of Banach algebras", Van -

a

Nostrand (1960);

Theorem 2.3.5(ii)).

By the preceding Proposition, there exists an isometric linear map w of t~ into Ma(S)*ILWUC(S)•

So for some closed subsapce P of Ma(S)*, we have -1

~

w(1) dense in PfLWUC(S)•

Hence the inverse map w

~

(defined on w(t )) ~

extends to a unique isometric linear map T of P/LWUC(S) onto t . The dual map T* : (1~)* ~ (P/LWUC(S))* is isometric, linear and onto. But then A= LWUC(S)

.L

can be identified with the dual of Ma(S)*ILWUC(S)•

So each

element of (P/LWUC(S))* can be identified with the restriction of some element of A to P. This completes our proof (on noting that (1~)* is nonseparable). Let S be any topological semigroup satisfying the hypothesis of Theorem 2.11 and note that WUC(S)* is a Banach algebra under an Arens operation given by ae'a(f) := a(aof) for all a,a

£

WUC(S)*;

and f

£

aof(x) := a(xf), WUC(S) and x

£

S.

Now, by Theorem 2.11,

card(IM(WUC(S))) ~ 2c. Let B := {~ £ WUC(S)* : ~(1) = 0 and w(xf) = w(f), for all x £ S and f £ WUC(S)}. Note that B is a right ideal of WUC(S)* such that B•'B

= {0}. By the result referred to Rickart's book in the proof

of the preceding Theorem, we have that the radical of the algebra WUC(S)* contains B. Now fix ~ 0 £ IM(WUC(S)) and note that IM(WUC(S)) - ~ 0 is c contained in B, and so card(B) ~ 2 • Consequently the radical of WUC(S)* is nonseparable.

109

PROBLEM. Let S be a cancellative infinite foundation semigroup. We conjecture that the radical of the algebra WUC(S)* is nonseparable. (The preceding remarks say that, to a large extent, the conjecture is true if there exists a left invariant mean on WUC(S) and S is not compact. Even for groups, this conjecture is still open [52j.) REFERENCES. of L~(G)*.

Let G be a locally compact topological group and R the radical Civin and Yood [19] showed that R is infinite dimensional for

G nondiscrete and abelian or G • z. E.E. Granirer [52] proved that R is nonseparable if G is amenable or nondiscrete. Our theorem above is inspired by the latter paper of Granirer. In [56] Gulick showed that the quotient space L~(G)fc(G) is nonseparable if G is not extremely disconnected. results on the radical of L~(G) or UC(G)* can be found in [16].

110

Other

5 Characterizations of absolutely continuous measures This chapter forms a natural continuation of Chapter 2 bearing in mind our experiences in Chapter 3.

The message of Chapter 2 is that one can study

various spaces of measures, on a topological semigroup, that can be different in general but all coinciding with the group algebra in the case of a locally compact topological group;

and further one such space, namely the

space of measures which are absolutely continuous, seems to have many interesting properties.

Chapter 3 then presents a very large class of

topological semigroups, the so called stips, which have attractive topological "homogeneous" properties and include the class of all locally compact topological groups as a "very special" case. In this chapter we shall study various characterizations of the absolutely continuous measures on a stip and also include some results very close to characterizing such measures.

Throughout this chapteP S denotes a stip unless othePwise explicitly stated. Blanket Assumption.

1. CONTINUITY OF MEASURES UNDER TRANSLATION ON A STIP Noting that the map

(x,~) ~ x*~

of S x M(S) into M(S) defines a left action

of Son M(S), the following result follows as a special case of Theorem 3.4.6.

(If S is also the foundation of M (S), then one may also deduce our a next Theorem from Corollary 2.2.4.) 1.1 THEOREM.

If

~ ~

M(S), the following items aPe equivalent:

the map 1 is weakly continuous; ~ (ii) the map 1 is noPm continuous. (i)

~

Let p E Ma (S) and V be a neighboUPhood of 1. -isolated idempotent e in V such that

1.2 COROLLARY.

exists a

<5

supp(~)

Then thePe

£ eSe.

111

Proof.

We may assume p to be positive and find a a-compact set M such

IIPII •

that

p(M).

Vn := {x e S Since A :=

Recalling Lemma 2.1.4, our preceding Theorem says that

lli*p - p II

<

~} is an open neighbourhood of 1 for all n e lN.

s1 is dense inS (by Theorem 3.2.2 ),we can find a countable set {xn e Vn n s 1 : n e lN}. By Lemma 3.3.4 we can find e e: V n ES~ such

that AM C eSe.

and so p(S\AM)

Now by definition of Vn we have

= lim

in*p(S\AM)

=0

since each in*P is concentrated on AM.

n

Hence p is concentrated on AM. supp(p)

Since eSe is c1osed we conclude that

S AM S eSe.

If S is a foundation semigroup then, with Exercise 3.3.7(i) in mind, one can easily note that

1.3 REMARK.

Recalling Exercise 3.3.7(iv) we note the distinction between the foundation of Ma(S) and the union of the supports of measures in Ma(S). notions coincide in the case of a group.]

[The two

For a general locally compact topological semigroup H, Exercise 2.1.15 (f) teaches us that we may have measures v,p e: M(H) such that p e Mn(H) and v

<<

IPI

but v ~ Mn(H).

Contrary to this, for a stip S we have M (S) n

= Ma (S)

and so M (S) is solid. n

Partly towards this end we have the following result.

Let A1 (S) :• {n e: M(S) tn is weakZy continuous at 1}, Then v e: A1 (S). Al(S) and v E M(S) with v << IPI·

1.4 LEMMA. p E

Proof. £

Evidently A1 (s) is a norm-closed linear subspace of M(S). > 0 and f e: C (S), we note that 0

and therefore U := {x e: S 112

Given

is a neighbourhood of 1. Since p

£

A1 (S), if h

£

M(S)* with llhll = 1 then

V :• {x £ S : lh(f.(p~*p)) I < £}

is a neighbourhood of 1.

u n v~

{x

€

s:

SoU n V is a neighbourhood of 1.

lh(f.p) - h(x*(f.p))l < 2£} •

Hence f.p € A1 (s). Since {f.p: f follows that v £ A1 (S).

If

1.5 THEOREM.

p £

C0 (S)} is dense in L(S,IPI>. it

£

M(S), then the foZZowing items

(i) 1 p

is

~eakZy

continuous at the point 1;

(ii) 11PI

is

~akZy

continuous at the point 1;

is

~eakZy

continuous on

(iii) 1P

But

~

equivaZent:

s.

Proof. Evidently (iii) implies (i). The equivalence of (i) and (ii) is a special case of Lemma 1.4. So to complete the proof we assume item (i) and prove (iii). In view of Lemma 1.4 we may assume that pis a positive measure. Let C be a compact subset and (xa ) a net converging to x in S. There exists a subnet (ya) of (xa) such that lim Yo*P(C) • lim x *p(C).

a

a

p

(1)

a

By the Hahn-Banach theorem, one can find a real-valued functional h such that, for all real-valued v £ M(S), we have

£

M(S)*

(2)

So, in particular (3)

Let

£

>

U :• {u

0 be given. £

Then item (i) implies that

-1 C)-p(x-1 C) S : lu*p(x

I

< £}

n {u

£

S

113

is a neighbourhood of_ 1. Since s 1 is dense inS we can find u € int(U) n s 1 • Now (xU)u-l is a neighbourhood of x, by Theorem 3.2.6(i). So eventually -1 (y 6 ) S (xU)u • Hence there is a net (u 6) c U such that, eventually (4)

Now i•\.I(C)

~

lim xuB*\.I(C) + £

= lim

yBu*\.I(C) + £

by definition of u by item (4)

B ~ h(u*\.1> + £ ~

by definition of u

h(J.I) + 2£

= lim B = lim

Ya*\.I(C) + 2£

x *J.I(C) < lim x *\.I(C) -(I

-

(I

(I

(I

~ X*\.I(C) + 2£

Consequently \.1

by item (2)

€

+ 2£

by item (3) by item (1)

+ 2£

by Lemma 1.4.l(ii).

M!(s) and item (iii) follows, by Lemma 2.1.4.

1.6. Swrrnarising CoroUarry 2.2.4(i), Theorems 1.1 and 1..';, fo:t' each \.1 € M(S) ~e have the foZZ~ing items equivalent: (i) \.1

€

M!<s>;

(ii) \.1

€

M1 (S);

(iii) 1\.11

n

€

M!<s>;

(iv) R.\.1 is

~akty

continuous at the point 1.

1.7. Theorem 1.1 was first proved by Sleijpen [94] in a way relying heavily on measure theoretic techniques. The proof given here is due to The other results of this section are taken from Dzinotyiweyi [34). Sleijpen ( [94] and [92]).

114

2. MEASURES WITH SEPARABLE ORBITS Recalling Definitions 3.4.1 we consider weakly and norm separable left orbits of measures in M(S) in terms of the left action of S on M(S) given by (x,~) ~ i•~ for all x £Sand~£ M(S). First we have the following consequence of Theorem 3.4.4 and item 1.6. 2 .1 THEOREM.

£

M(S). M1 (s) for a

£

s1 ;

(ii) If~ Proof.

and~£

If~ has a weakLy separabLe Left orbit over U, then x*~

(i)

aU x

Let U be a compact neighbourhood of 1

Let

£

M!(s), then~ has a norm separabLe Left orbit over E >

O(y) := {s

u.

0 be given, y £ S fixed and consider the set £

S

To prove item (i) it is sufficient to show that O(y) is closed and Theorem 3.4.4 will imply the result.

To the latter end we first note that the

continuity of the function s ~ s*~(f) defined on S (f £ C(S)) implies that

is closed in S.

Since M(S) is the first continuous dual of C (S), it is 0

trivial to note that, from

we have that

Thus O(y) is closed and item (i) follows. Now item 1.6 says that ~ £ M!(s) implies that {i*~ compact and so item (ii) follows. 2.2 Notes.

x £ U}

is'norm

The subject of absolute continuity of measures with separable

orbits particularly on locally compact groups, has been studied quite extensively. This seems to have started with a paper of R. Larsen [7oJ, and subsequently various other papers appeared- see e.g. Tam [100]; T.S. Liu, A. van Rooij and J.K. Wang [73], G.L.G. Sleijpen [93] 115

and H.A.M. Dzinotyiweyi ( [33] and [34]). In this chapter, Theorem 2.l(i)", (or rather its general form: Theorem 3.4.4) has enormous applications.

In fact almost every Theorem we prove in

this chapter can be traced back to have its roots in Theorem 2.l(i). Theorem 2.l(i) was first proved for stips admitting a certain measure theoretic condition in [93] and its proof in the general form given above first appeared in [33] • 2.3 Remark.

In general there are stips (even foundation semigroups with

identity element) admitting measures having norm separable left orbits but are not absolutely continuous.

See for instance Example 4.4.

In preparation for our next Theorem we give a characterization of ~(S) in the following Lemma. 2.4 LEMMA.

The fottowing items

a~

equivatent foP

any~ E

M(S):

(i) ~ E M!(S); (ii)

foP each 6-isotated idempotent e of s, we have e*~*e EM (eSe). a

Proof.

Evidently (i) implies (ii).

We now assume (ii) and prove (i).

Let

U be an open relatively compact neighbourhood in S and A any countable sub-

U n s 1 • By (the right handed form of) Lemma 3.3.4 we can find a 6-isolated idempotent e E S such that

set of

A(suppl~l u {1}) SeSe

and hence

So item (ii) implies that {a*e*~*e : a E A} is rela:ively norm compact, by the compactness of

U n eSe.

for all countable subsets A of {x*~ : x E

U n s1 }

Thus {a*~ : a E A} is relatively norm compact

U n s1 •

Hence, we have that

is relatively norm compact.

Since each norm-compact subset of M(S) is weak*-closed and each weak*closed subset of M(S) is norm-closed, the fact that the weak*-closure of {x*~ x E U n s 1 } is {x*p : x E U} implies that the latter set is norm 116

compact. Hence i U- is norm continuous, by Lemma 2.2.1 and item (i) llt follows, by item 1.6. 2.5 DEFINITIONS.

Recalling the definition of equi-absolute continuity just

before Lemma 2.1.6, we now formally define spaces of such measures. M!q-a(S) := {p E M(S) : x*IPI

<<

M~q-a(S) :a {ll E M(S) := IPI*x

lvi*IPI

<<

for all

X

lvi*IPI for all x

Let

E supp(v) and v

E

M(S)}

supp(v) and v E M(S)}

E

M (S) := M1 (S) n Mr eq-a eq-a eq-a Here S is any C-distinguished topological semigroup. It would be interesting to investigate the properties of the space of equi-absolutely continuous measures.

In the following example we show that

the space may differ from that of absolutely continuous measures for a general locally compact topological semigroup • 2.6 EXAMPLE.

Let G2 :=

.

rr z3 be the product topological group of countably

i=l

many copies of the (discrete) cyclic group of 3 elements H := G3 u {e}u{(n,x) E :N x G3 : elements} with the operatiop

the set {mElN :x(m)

z3

+ 0}

:= {0,1,2}.

Let

has at most n

y(n,x) = (n,x)y := xy, ab=ze=ez:= e (y£ G3 ; a,b,(n,x) E :NxG 3 and z E H) Then, with e isolated, H is a locally compact topological semigroup. 00

product measure ll :•

rr

(IO

+

ll)

i=l

The

defined on G3 (and hence on H) can be

shown to be equi-absolutely continuous but not absolutely continuous. For a stip, we show in our next Theorem that equi-absolutely continuous measures are precisely the absolutely continuous measures. Towards this end we first show that M1 (S) is solid. eq-a 2. 7 LEMMA.

1 Let ll E Meq-a (S) and n

E

M(S) ~th n

<<

IPI·

Then n

E

1 Meq-a (S).

117

Proof.

Let v

E

M(S) be fixed.

Since lnl

<<

IPI ,

we evidently have (1)

By the Hahn-decomposition Theorem, we can find non-negative measures p1 and p 2 in M(S) such that

Recalling (1), we see that (2)

We can find a Borel subset F of S such that p 1 is concentrated on F and p 2 (F) = 0. Now suppose that, for some Borel E c S, we have lvl*lni(E) = 0.

Then

So

since

p E

1 Meq-a (S).

Recalling that lnl x*lni(E n

F) • 0

IPI

<<

we get

for all

X E

supp(v).

(3)

Now clearly supp(x*lnl> S supp(lvl*lnl>. for all x E supp(v). Hence, since lvl*lnl is concentrated on F (by (2)), we have that x*lnl is concentrated on F, for all X E supp(v). This together with (3) says that x*lni(E)- 0 and hence that for all x E supp(v). This completes our proof.

118

If

2. 8 THEOREM.

(i)

~ £

M(S) then the

fo~~~ng

items aPe

equiva~ent:

~

(ii) ~

Proof.

That (ii) implies (i) follows from Lemma 2.1.6.

and prove (ii).

We now assume (i)

In view of Lemma 2.7 and the fact that M!(s) is now a

closed solid space, we assume that

~

is a positive measure with compact

support K (say). First suppose that 1 is 6-isolated. 1.

Let V be a compact neighbourhood of

Proposition 3.3.8 says that there exists a countable subset B of

A:= VKuV and a G6 compact subgroup G contained in F := int(V), such that items (i) to (iii) of the Proposition hold. Let {bk} be an enumeration of BnGV and v be the normalised Haar measure on G.

By Proposition 3.3.8(ii) we have that

Hence (x

£A).

(1)

CD

L et

(i).

2-k_.. .....h_ _k

v := ..'<' k=l Since ~

£

and note that supp(v) • GB = GV, by Proposition 3.3.8

M1 (S), we thus have that eq-a for all x

£

GV.

Consequently x*v*~ << v*~ for all X £ GV.

(2)

Now Proposition 3.3.8(iii) and the fact that C(S) is dense in L 1 (v*~), imply that {f £ L 1 (v*~) : x*v(f) = f(x) for v*~-almost all X£ GA} is a separable subset of L1 (v*~). {x*v*~ : x

£

Hence, by (1) and (2), we have that

GV} is separable.

By Theorem 2.l(i), we have that -

x*v*~ £

1 Ma(S)

for all

X£

S1 •

(3)

119

~-isolated,

Since 1 is

subset C :• {en : n

£

Theorem 3.3.6 implies the existence of a countable

1N} of

s1

with 1 ~::

So 1 ~:: CG.

C.

CD

Taking v 0 :=

I

2-ncn*~ we have 1 ~::

CG

n=l

= supp(v 0 ) and so

It is trivial to note that p ~:: M1 (S) eq-a implies that e*p*e € M (eSe) for all e € ES. By the above argument and eq-a Lemma 2.4 we thus have Now suppose 1 is not

~-isolated.

This completes our proof.

[98] .

The proof of the preceding Theorem is taken from Sleijpen

The

Theorem is essentially a consequence of Theorem 2.l(i) which uses the notion of measure with separable orbits.

It remains an open problem whether one

can supply a proof independent of the notion of separable orbits.

Example

2.6 is taken from [96]. 3. MEASURES VANISHING ON EMACIATED SETS 3.1 DEFINITIONS.

A subset A of S is said to be Zeft-emaciated if AA-l is

not a neighbourhood of the identity element nf S.

The collection of all

G~

left-emaciated subsets of S is denoted by Le and we define E~(S) := {p € M(S) : p(F) • 0 for all F € L }.

compact

e

Na •

We denote the collection of all M (S)-negZigibZe subsets of S by a

Each compact Left-emaciated subset, F, of S is both meagpe and Ma(S)-negZigibZe. 3.2 PROPOSITION.

Proof.

That F is Ma(S)-negligible follows from Lemma 2.1.9 and our definitions. To see that F is meagre • let B : = S\ FF -l and note that 1

£

B and BF n F = 41·

If x £ F then x € Bx and Bx n F • 41 1 which shows that int(F) • 41· is closed, we have that F is meagre. 120

Since F

The proof for the following Proposition is easy and left as an exercise for the reader.

For att v € M(S) and p € EM1 (S) ~th v FUrther EM1 (S) is norm closed in M(S).

3.3 PROPOSITION.

that

v € EM1 (S).

<<

IPI, ~e have

We now begin an argument culminating in the result that every measure in EM1 (S) is left equi-absolutely continuous.

We partition part of our proof

into some lemmas. 3.4 LEMMA. Let p € M(S) + , B c S and K a compact subset of S. each x € B, v € S and compact-F c s ~th F S x-1K\B- 1K the set {t

€

Then for

s

Proof. There exists a net (xa) in B converging to x. For each t € s 1 and compact neighbourhood T of t, we have that ((xvT)t-1)v-l = (xvT)(vt)-l is a neighbourhood of x, by Theorem 3.2.6(i).

So there is a net (t) in T a

converging to t such that

Now since F :• x

-1

K\B

-1

K an easy manipulation shows that

+

for all a.

(1)

Let W :• {s € S : v*s*P(F) > !}, for all n € 1N. By the upper semin -n continuity of the maps~ s*~(v- 1 F) (recall Lemma 1.4.1), we have that W

n

is closed. From (1) and Lemma 1.4.l(ii), we have that

Since twas arbitrarily chosen in

s1 , s1

it follows that wn is nowhere dense. Consequently {t

€

S : v*t*p(F) ~

0} =

.u

is dense in S and Wn is closed,

Wn is meagre.

n•l 121

3.5.

In this item we develop an argument that is needed in the following

two Lemmas. Let K,F and V0 be compact subsets of S such that K is a is a neigbourhood of 1.

...

G~-subset

and V0

Thus there is a decreasing sequence {Un} of open

n un • n-=1

subsets such that K •

Remembering Theorem 3.2.6, we can find a sequence {Vn} of open neighbourhoods of 1 and a sequence {vn} of elements in vl

£

s1

such that

vo. vn £ vn

2 c vn+l -

v- 1v n vn n n

...

Then one can easily note that H is a compact c 6-subn Vn. n•l semigroup of S with identity element 1. We then get

Let H :=

H-l(HK) n F •

...

K) n F) c n (V-l(V n n

n=l

...

n un • K

n=l

and hence H-l(HK) n F = K n F

(1)

Similarly taking V F in place of F in (1) we get 0

H-l(HK) n V F = K n V F 0

(2)

0

Let e be an idempotent in the minimal ideal of H.

Then from Theorem

1.2.3, we have that G := eHe is a compact subgroup of S with identity element e. For all h £ H, we have that H n F £ h-l(H-l(HK) n V0 F) n F C h-lK n F

by (2)

~ H-l(HK) n F = K n F

by (1).

Consequently K n F • h-lK n F 122

for all

h £H.

(3)

3.6 LEMMA. Let K be a compact G6-subset of S and p € M(S)+ with M :• supp(p) compact. Let x € cto{y € S : y*p(K) = 0}. Then the~ exists a B c s and an e € s such that

(ii) ex €

B;

(iii> ex*p = x*p. Proof. Let X be a compact neighbourhood of x and take F :• XM in the argument presented in item 3.5. Assuming the notation of 3.5, let P :• {y € eX : y*p(K) = 0}. By equation (3) of item 3.5, we have ex € P. 1 (v ex) n P, for all n € 'IN. Hence we can choose an € vLet n n B :={an: n € D} and B := H-1 (HB ). 1

1

the latter set is clearly an F0 -set.

p(B~lH-l(HK)) <

Now

CD

an*p(H-l(HK))

I

n•l CD

a n *p(K n Vo F)

by 3.5

(2)

• o. -1

-1

Thus B K is contained in a set of p-measure zero, and so B K is p-measurable and ~(B- 1K) • 0. This proves (i). To prove (ii), let U be an open neighbourhood of ex. Recalling the notation of 3.5, we claim that there exists an n € 1N such that -1 (eVne) (eV- ex) n eX c GU. n -

(1)

To verify this claim we first note that CD

-

n (eVne) n=l

-1

-

(eVn(ex)) n

·eX~

-1

G (Gex) n eX c e

-1

Gx.

So CD

-1 (eVnex) n · ( ( (eVne) n eX)\GU) =

n•l

+• 123

Since G is a group GU is open, and so the above represents an intersection of closed subsets of the compact space eX.

Hence for some fixed n

£

we

~

must have n

-

n (((eVke) k=l

-1

+•

(eVkex) n eX)\GU)

Consequently item (1) follows, since {V } is a contracting sequence. n

Now an easy exercise on our definitions shows that V- 1 (v ex) n e X c ( eV e )- 1 ( eV ex ) n e X• n n n n From (1) and (2) we have that B1 n FU Bn U~

+·

It follows that ex £

~

Band

(2)

+• (ii) is proved.

Finally, from equation (3) of 3.5 item (iii) follows.

3.7 LEMMA. Let K be a compact G6-subset of M := supp(~) is compact and {y £ S : vy*~(K) Then there e~ists a c c s such that (i) C-lK is ~-measurabLe and p(C-1K) (ii) 1

Proof.

Sand~ £ ~

0} is

M(S) + such that

meagre~

for each v

£

S.

0;

c.

£

Let U be a compact neighbourhood of 1 and take F := UM in the

argument presented in item 3.5. a finite subset of U n {y £ S : -1

U c Vn+ 2 (Vn+ 2An)

Assuming the notation of 3.5, let An be vn+ly*~(K)

for all n

£

= 0} such that

'IN.

GO

Let

c1 :• u v +lA and C := H-1 (HC 1 ). n=l

n

n

Then arguing as in the proof of

Lemma 3.6(i) one easily obtains our item (i). To prove (ii), let 0 be any open neighbourhood of 1 with 0 cu. By -1 Theorem 3.2.6(i) we can find u £ 0 n s 1 such that (HO)u is a neighbourhood 1 of Hand therefore we can find n £ 1N with V c (HO)u- • Since u

-1

£

124

n -

U c Vn+ 2 (Vn+ 2An)' there exists an a£ An such that

Consequently C n 0 ~ $• and so 1 £

C.

Now we can prove our first theorem in this section. 3. 8 THEOREM.

Proof.

We have

Let~£

that

t

EM (S).

3.3, we shall assume that

~

To show that~£ Mt (S), in view of Proposition eq-a is positive and M := supp(~) is compact.

Let v £ M(S) and K be a compact c 6-subset of S such that lvi*~(K) Thus

0

=

f y*~(K)dlvl(y),

y*~(K) = 0

so

= 0.

for v-almost every y £ supp(v).

Fix x £ supp(v) and note that x £ clo{y £ S : y*~(K) = 0}. Now let B and e be chosen as in Lemma 3.6.

We therefore have that

0 = x*~(K) ~((ex)-lK\B-lK) if and only if ~(F) = 0 for every compact c 6-set F contained in (ex)-1K\B-1K. -1

-1

Let F be a compact G6-set with F c (ex) K\B K. {t £ S : v*t*~(F) ~ 0} is meagre, for each v £ S.

Then Lemma 3.4 says that

Next Lemma 3.7 says there exists a C c S such that

~ Hence F\C ~ F is ~-measurable and 0 = ~(F) = ~(F\C F) if and only if ~(A) = 0 for each compact G6-set A for which A c F\C- 1F. But that A c F\C-lF implies

that AA-l n C

= $.

Since 1 £

Cwe

thus have A £ Le and so ~(A)

=0

by our

definition of EMt(S). ~

£

Hence x*~(K) Mt (S). eq-a

=0

and so

x*~ << lvl~. for all X£ supp(v).

Thus

Combining Proposition 3.2 with Theorems 3.8 and 2.8 we have the following theorem. 125

For

3. 9 THEOREM. (i)

~

E

~

E

every~ E

M(S), the

foZZ~ing

items are equiuatent:

M!(S);

1 Meq-a (S); (iii) ~ E EM1 (S). (ii)

a consequence we now see that if a measure is singular with respect

As

to every measure in Ma (S), then such a measure is concentrated on an Ma (S)negligible subset of S.

(This is implied by Lemma 2.1.9 and Corollary

3.10). Let~

3.10 COROLLARY.

E M(S) be

such~

1

Then there e:cists a

L Ma(S).

a-compact subset A of s such that

Proof. M :=

Without lossing generality we may assume that

supp(~)

~

is positive and

is compact.

There exists a sequence {Kn} in Le such that .. m (1) ~( u K ) = supp{~( u F ) n=l n n=l n

..

Setting K :• u Kn' we see from (1) that n=l ~-~IKE

1

Ma(S), by Theorem 3.9.

It

~-~IK

follows that

1

E EM (S) and hence that ~

=

~I

K since

~

1

.L Ma (S) and

Hence

· u~ 11

-

~.

From Lemma 3.4, for each v E S, we have that {t E S : .v*t*~(K) ~ 0} •

.. U

n=l is meagre. Hence, by Lemma 3.7, there exists a subset C such that 1 E C-lK is ~--measurable and p(C-lK) • O. We can then find a a-compact subset A of D := K \ ~(A) • p(D) • ~(K) =

126

~~~~~·

c-1K such

that

C,

That A C K \ C-lK implies that AA-l n C 1

-

+ int(AA

-1

3.11 Note.

).

= t•

Since 1 E

C,

it follows that

This completes our proof.

In the case where S is a locally compact group, the result

that EM1 (S) = M!(s) was proved by S.M. Simons [91]. case of stips is due to G.L.G. Sleijpen.

The extension to the

The results given in this section

are taken from Sleijpen [93] • It is trivial to note that in the case where S is a foundation semigroup, Borel

ema~iated

sets are Ma(S)-negligible.

is in Ma(S) as soon as

=0

~(F)

In that case, a measure

~

E M(S)

for every compact Ma(S)-negligible subset F

of S. 4. LOCALLY QUASI-INVARIANT MEASURES We now resume studies introduced in Section 4 of Chapter 2.

The results of

this section are taken from [93] and [92] • 4.1 DEFINITION.

A measure m e M(S) is said to be

Ze~

ZocaZZy quasi-

invaPiant if there exists a neighbourhood U of 1 such that x*m

<<

lml

for all

X

E U.

Recalling item 2.4.5, the reader can easily note that the notion of local quasi-invariance is different from that of equi-quasi-invariance introduced in Chapter 2. However the technique employed in the proof of Theorem 2.4.2 is similar to that employed in the following Theorem. 4 • 2 THEOREM.

We have that

= u{L(S,m)

Ma(S)

: mE Ma{S) + and m is left locally quasi-invariant}.

Proof. Let U be a compact neighbourhood of 1 an~~ E Ma(S). sufficient to prove the existence of an m E Ma (S) such that ~ x*m«m (x e U). compact {p e ~). {xp,n }n e 1N

{x*l~l :

X

It is <<

m and

By LeDDna 2.1.4, we have that {x*l~l : x e uP} is weakly By Appendix B.4(iii), we can find a sequence

such that {xp,n *I~ I : n e 1N} is weakly dense in

E UP}.

Let

127

I l~,n
and note that

l.l <<

*

I I 2 - X p,n }.1

m and m ~ M (S)+. a

Now if m(E) = 0 for any Borel E c S we have x *Ill I (E) = 0 (n,p ~ :N) and .. p,n .. hence x*llli(E) 0 (x ~ u uP). Noting that XX ~ u uP for all p=l p,n p=l p,n ~ :N and

X

~

u we

conclude that x*m

<<

m (x ~ U).

It is nice to note that, using the notion of measures vanishing on emaciated sets, we have that left locally quasi-invariant measures are "very close" to being left absolutely continuous. 4.3 THEOREM.

Let m

x*m ~ M!(S) Proof.

~

M(S) be Z.eft Z.oaaUy quasi-invax>iant.

for a7,7,

Then

X~ S1 •

There exists a neighbourhood U of 1 such that y*m

<<

lml (y E U).

Let x ~ s 1 be fixed. Since S~ 1 x is a neighbourhood of 1 (by Theorem 3.2.6 (iii)), we can find a y ~ U n s 1 and avE s 1 with x vy. Let K

~

Le.

By noting that K n (S\KK

-1 -1 ) K =+and 1

Lemma 3.4 implies that {t ~ S : t*lmi(K)

>

~

0} is meagre.

-1 clo(S\KK ), Since yU-l is

a neighbourhood of 1, we can find at~ yU-l such that t*lmi(K) = 0. Choose u E U with tu = y. Then y*m(K) = ~*m(t- 1 K) s 0 and hence y*m ~ EMR.(S). By Theorem 3.9 we have y*m ~ M!(s).

Consequently x*m

In Theorem 4.3 one cannot replace s 1 by S. 1

·ConsiderS := {O}u { n: n ~ :N} with maximum operation and Thus S is a foundation semigroup with identity the usual topology.

4.4 EXAMPLE.

element.

Let m :•

ever since

0.

128

0

+

M!(s) and

..

I 2-n (~) and note that x*m n=l

0

<<

<<

m we have that m. M!(s).

m (x ~ S).

How-

5. CONTINUITY OVER DIRECT TRANSLATES OF COMPACT SETS Our definition of absolute continuity involves inverse translates of compacts, that is sets of the form x- 1K and Kx- 1 where K c S is compact and x

S.

E

In this section we wish to involve direct translates of compact sets,

that is sets of the form xK or Kx. A.C. Baker and

J.W.

Baker

[4]

studied the collection of measures ~

E

M(S)

such that x ~ l~l(xK) and x ~ I~I(Kx) are maps of S into lR continuous on S or at a given point of S; group.

where S is any locally compact topological semi-

Such spaces don't seem to be as nicely behaved as M (S), in general. a

For a stip we however have the following characterization of one such space as Mt(S). a

For

5.1 THEOREM.

every~ E

M(S), the foLLowing items are equivaLent:

(i) For every compact C c S the fUnction x continuous at 1; (ii) ~ Proof.

E

of S into C is

M!(S).

To show that (i) implies (ii) let K

E

Le be fixed and

£

> 0

be

There exists a compact neighbourhood U of 1 such that I~I(UK\K) <

given.

Taking B := S\KK-l we note that 1 all x

~ ~(xC)

E

E

B and bK

n K = $ for all b

E

B.

£,

So for

U n B we have that

l~(xK) I

=

l~(xK\K)I < I~I(UK\K) < £ ,

and hence, by (i), we get ~(K)

= 0 and~

E

EMt(S).

By Theorem 3.9 we have

(ii). Next we assume (ii) and prove (i). that

~

is a positive measure.

Since M!(s) is solid we may assume

(Our proof is similar to that of Theorem 1.5.)

Let C be a compact subset of S and (x ) a net converging to 1. a

There exists

a subnet (y 6) of (xa) such that lim

--a

~(x

a

C).

As in the proof of Theorem 1.5, let h real-valued v

E

(1)

E

M(S)* be chosen such that, for all

M(S) we have that

129

(2)

In particular h(~)

=lim

a

~
Let & > 0 be given.

(3)

Then item (ii) implies that

is a neighbourhood of 1. We can choose u

int(U) n s 1 and note that uU-l is neighbourhood of 1. So -1 • There exists a net(ua) s_ U such that eventually

£

eventually (ya) c uU

(4)

.. u.

Now from our definition of U, (4), (2), (3) and (1) (as in proof of Theorem 1.5) we have

<

-

Hence lim

lim

~(x

a

~(xaC)

=

a

C) + 2&

~(C)

~~(C)

+ 2&.

and (i) is proved.

a

5.2 EXAMPLE.

In general it is not true that x

~ ~(xC)

is a continuous map

for all ~ £ Ma(S) and compact C s_ s. For example take S := [o,lJ with the usual topology and multiplication given by xy := min(x+y,l)

for all

Then Sis clearly a stip and l

x,y £

£

Ma(S).

S. Let C := {0}, xn := 1 - ~ and note

that 0 = l(xnC) ~ l(lC) • 1. The results of this section are taken from G.L.G. Sleijpen

130

<(92]

and

(93]).

6. SOME FURTHER NOTES 6.1.

If the stip S is in fact a foundation semigroup then we recall from

Theorem 2.2.6 that Mt(S) = Mr(S). a a

In that case results of this chapter

remain true with Ma (S) in place of ~(S). a stips in general.

In fact this is also true for

We avoided giving this general form here for two reasons:

Firstly the proof as known in the general case is rather complicated (see

[94]), and secondly, as remarked in Chapter 3, a stip may after all be a foundation semigroup. THEOREM (Sleijpen (94J).

For a stip S and

1J E

M(S), we have the following

items equivalent: (i)

1J E

(ii) tll

(iii) r

1J

Ma(S);

is weakly continuous at 1; is weakly continuous at 1;

(iv) both t 6.2.

1J

andr

1J

are norm continuous on s.

We noted in Section 3 of Chapter 2 that the space of equi-regular

measures is far from coinciding with the space of absolutely continuous measures for a very large class of locally compact topological semigroups. With this in mind, the next Theorem shows that a stip behaves just as well as one would expect, following the results of the preceding sections. THEOREM (Dzinotyiweyi and Sleijpen

!39]).

For a stip S we have that

Ma(S) ,. Me(S). 6.3.

The reader who is interested in investigating whether every stip is a

foundation semigroup is referred to the following partial result. THEOREM (Sleijpen ( [92] and

[95])).

G6-subset or M(S) is semisimple.

Let S be a stip such that s 1 is a Then s is a foundation semigroup.

131

6 The convolution algebra

i 1 (S)

In this Chapter we prove certain results in which the algebra 11 (S), first introduced by Hewitt and Zuckerman [60], plays a pivotal role. We do not give a systematic study of 11 (s) as such. Most of the results we discuss have a significant algebraic flavour. First we formally define the algebra 11 (s).

Throughout this

Chapte~

tet s be a

disc~ete

topotogicat

semig~oup.

Let 11 (s) denote the set of all functions f in m(S) such that l:

lf(x) I

< ...

X€S With pointwise addition and scalar multiplication, with convolution f*g(x) : ..

f(u)g(v)

(x

€

: = t If <x> 1 1 (S) X€S

1.

l:

S)

u,v£S UV =X

as product and with the norm

II f II = II f II

1 1 (S) is a Banach algebra called the disc~ete semig~oup atgeb~ of S. One may identify 11 (s) with the measure algebra M(S) and we shall exploit this identity when convenient. 1. REGULARITY·OF MULTIPLICATION IN WEIGHTED SEMIGROUP ALGEBRAS As remarked in Section 5 of Chapter 3, the property of regularity of multiplication in Banach algebras is invariant under passage both to subalgebras and to quotient algebras.

In this section we study the regularity

of multiplication in algebras related to the discrete semigroup algebra. 1.1 DEFINITION.

By a

w : S + 1R such that 132

~eight

fUnction on S we mean a positive function

111(xy)

~

w(x)w(y)

(x,y

€

5).

The set t 1 (5,w) of all complex-valued functions f on 5 such that fw

€

11 (S) 1

is a Banach algebra with convolution as product and norm given by

11£11

:=

llfwll

•

1

1 (5)

If 111 is a measurable weight function on a locally compact topological group G, then the set L1 (G,w) of (equivalence classes of) all measurable complex-valued functions f on G such that fw

€

L1 (G), is a Banach algebra

with convolution f*g(x) := JG f(t) g(t- 1x)dm(t)

(x € G)

as product, (where m is a given left Haar measure on G), and with the norm

11£11

:=

llfwll

•

1

L (G)

Corresponding to a weight function w on S we define the function 5

X

n on

5 by

w(xy) n(x,y) ::o w(x) w(y) • We are now in a position to establish a criterion for the regularity of multiplication in the semigroup algebra 11 (s.w).

Then (i) implies (ii), and Let w be a weight function on s. if 5 is cancellative (i) and (ii) are equivalent. ~heFe 1. 2 THEOREM.

(i) 1 1 (S,w)

has iFFegular multiplication

theFe exist sequences {xn}• {ym} of distinct elements of S such that the double sequence {n(xn ,ym)} has a non-ze~ Fepeated limit. (ii)

Proof. First, one can easily note that f € 11 (s) if and only if fw-l € 1 1 (S,w) 1 and h € 1m(S) if and only if hw € 1m(S,w). Now suppose item (i) holds. can find sequences

{fn }

and

By the argument preceding Lemma 4.5.1, we

• 1( ) { im } 1n 1 S

and h

€

1

m

(S) such that

133

a := lim lim hw(fnw n m

-1

*gw

*~w

-1

)

and b := lim lim hw(fnw m n

-1

*~w

-1

)

Now for any f,g E t 1 (S) we have

exist and a r/o b. hw(fw

-1

-1

)

=

f(x)g(y)O(x,y)h(xy)

E

x,yES and so if we define F : S x S

~

C by

F(x,y) :• O(x,y)h(xy) we have, in the notation of Appendix B.S, the map F' given by f(x)g(y)O(x,y)h(xy)

F' (f,g) :=

hw(fw

-1

*gw

-1

).

Hence by Appendices B.lO and B.7, we can find sequences {x } and {y } inS n

m

such that a 1 := lim lim F(x ,y ) n m n m

and ·b 1 := lim lim F(xn,ym) m n

+

exist and a 1 b1 • So (passing to subsequences if necessary) we may assume that the xn's and ym's are distinct, and that both repeated limits of O(x ,y ) and of h(x ,y ) exist. At least one repeated limit of O(x ,y ) must n m n m n m be non-zero (since a 1 b 1 ), and item (ii) follows.

+

Next we assume that S is cancellative and (ii) holds. lim lim O(xn•Ym) n m

>

C

>

We may assume that

0.

By taking a subsequence of {xn} if necessary, we further assume that lim

(x~ ,ym) > C

(n E lN).

m

Now arguing in a manner related but simpler than the proof of Lemma 4.3.1 we can find subsequences {un} of {xn} and {vm} of {ym} such that {unvm : n -< m} n {unm v : n

134

>

m} •

+

Let h be the characteristic function of the set {u v : n < m}. Then nm 1 1 1 hw £ 1m(S,w) and w- }, {; w- } are bounded sequences in 1 (s,w) such that n m

{u

-

-1 -

-1

Sl(u ,v ) > n m

={

hw(un w *vmw )

c

0

if n < m

if n > m.

By the remarks preceding Lemma 4.5.1, it follows that multiplication in 11 (S,w) is irregular. This completes our proof. Next we mention some examples of weighted discrete semigroup algebras with regular multiplication.

1.3 COROLLARY. (i) FoP a > 0 the fUnction w -

a

w (n) := (1 + lnl)a 1

1

Z

by

(n £ Z)

a

is a weight fUnction.

defined on the additive gPoup

(Z,wa) has pegulaP multiplication if and only if

a > 0.

(ii) Let w be a weight fUnction on the additive semigPoup of non-negative integePs z+, such that

1 . w(n+l) 1m w(n)

=

O.

n--

Then 11 (Z+ ,w) has PegulaP multiplication. Proof. (i) It is trivial to note that wa is a weight function and that 1 1 (Z,w0 ) = 11 (Z) has irregular multiplication, by Theorem 4.5.4. Now for any n

£

1N and a > 0 we have

so that each repeated limit of {Sla (n,m)} n,m is zero. result follows. (ii) For any fixed n . 11m

m--

(

1

£ ~

n n,m> = w

By Theorem 1.2 our

we have that

r

w(n+m-1) w(m+1)l • 0 1. l.o) ( n+m) __ 1m Lw . w(n+m-2) ••• w(m) J .

m

135

Hence every repeated limit of Theorem 1. 2. 1.4 REMARK.

~

is zero and our result follows, by

Indeed one can construct weight functions on Z+ which have the

property mentioned in Corollary 1.3(ii). w(n) :• e

-n2

For instance w given by

+

(n E lr )

is such a weight function. The examples discussed above refer to countable semigroups.

For such a

class of semigroups we have the following general result.

EVePy countable

1.5 COROLLARY.

semig~up

s admits a weight fUnction w

such that t 1 (s,w) has Pegular multiplication. Proof.

Let F be the semigroup freely generated by a countable set

{~ : k E :JN}.

X := ak • • ·~

1

Recalling that every x E F can be written uniquely as we define w : F -+ lR+ by setting

r

w(x) := 1 + k 1 + ••• + kr. Then w is a weight function on F such that w(xy) • w(x) + w(y) - 1

(x,y

E

F).

Clearly w(x ) -+ = for any sequence {x } of distinct elements of F. So if n n {xn } and {ym} are sequences of distinct elements in F, we have that w(xn ) + w(v·m) - 1 lim lim ~(x ,y ) lim lim 0 n m n-+<» m-+<» n-+<» m-+<» and si.milarly lim lim

~(x

n

,ym)

= 0.

By Theorem 1.2, it follows that t 1 (F,w) has regular multiplication. Since S is countable, there is an epimorphism a : F-+ S. We now define w'

S -+ lR by

w'(x) :• min{w(g) : g

E

F

and

a(g) • x}.

Then clearly w' is a positive function. 136

For any x,y

S we can find g 1 ,g 2

£

£

F such that

So

and taking the minimum over all such g 1 and g 2 we get w' (xy)

~

w' (x) w' (y).

Thus w' is a weight function on S. If

{x~}

and

{y~}

are sequences of distinct elements in S, we can find

corresponding sequences of distinct elements {x } and {y } in F such that n m w'(x') • w(x) and w'(y') • w(y ). So defining O' with respect tow' as n n m m done for 0 with respect to w, we have that O'(x',y') n

m

= O(xn ,y) m

~

0 as n,m

~ m.

Consequently 11 (s,w) has regular multiplication, by Theorem 1.2.

If G is a locatl.y compact topological equivalent:

1. 6 THEOREM.

items

~

(i)

group~

the fotl.OIJJing

G admits a weight jUnction w such that L1 (G,w) has ~guLar

mu "Ltip "Lication; (ii)

G is disc~te and countable.

Proof. That (ii) implies (i) is contained in Corollary 1.5. We now assume that (i) holds and prove (ii). Note that for any x,y £ G we have (1)

and G •

u Vk, where vk : • {y k .. l

£

1

G : k

~

w(y

-1 -1

)

w(y)

-1

}•

137

Now suppose we have the case where G is discrete and take, on the contrary, G to be uncountable.

Then, for some fixed k

£

:6, we have Vk

uncountable.

Thus we can find a sequence {ym} of distinct elements of vk. So if {xn} is any sequence in G, we have that 1

-k -< Sl(xn ,ym)

for any n,m

£

by (1).

1N;

Recalling Theorem 1.2, we have t 1 (G,~) irregular- contrary to assertion (i). By this contradiction we must have G countable. Next, suppose we have the case where G is non-discrete • G

=

...

u Vk, there is a k k=l

£

Since

1N such that Vk has positive Haar measure.

this k be fixed and choose q

Let

d(Vk), and note that

£

By Lemma 4.5.2, we can choose sequences {Cn} and {Dm} of compact sets with positive Haar measure such that

CD n C.D. • n m 1. J

~

for n < m and i > j.

So

C (D q) n C.(D.q) n m 1. J Taking

~

~

(CD n C.D.)q n m 1. J

~for

n < m and i > j.

to be Haar measure on G, let g

...

m

: .. ~(D q)

-1

m

Xnmq

and

u CnDmq. where A := u m=l n>m

Then

{fn~-l} and <~m~-l} are bounded sequences in L 1 (G,~) and h~

From the definition of Vk and item (1), we have that Sl(x,y)

138

~

k1

for all

x

£

G,

y

£

Dmq

and

m

£

1N.

£

L 1 (G,~)*.

Consequently

fD

I en

O(x,y)h(xy)d~(x)d~(y) =

{

mq

a>! ·- k

ifn > m

0

if n

< m.

Recalling the remarks preceding Lemma 4.5.1, we have that L1 (G,w) has irregular multiplication.

By this conflict we cannot have G non-discrete.

This completes our proof. 1.6 Notes.

The results of this section are taken from the paper of Craw

and Young [23].

There does not seem to be much work done on weighted group

or semigroup algebras. Recently W.G. Bade and H.G. Dales

I)]

have proved some results on weighted

semigroup algebras defined on the underlying semigroup [o,~).

We feel it

would be a useful contribution to extend some of their results to the weighted algebras t 1 (S,w) and L1 (G,w). 2. EXISTENCE OF APPROXIMATE IDENTITIES In this section we establish criteria for the existence of an approximate identity of the discrete semigroup algebra of an inverse semigroup. 2.1 DEFINITION. s

£

An

inverse semigroup is a semigroup S such that for each

S there exists a unique s' ss's

=s

2.2 EXERCISE.

and

s'ss

£

S with

=s

If S is an inverse semigroup then ES is a commutative sub-

semigroup. We refer the reader to the book of Clifford and Preston

[17] for more details

on inverse semigroups. Since elements of t 1 (s) are arbitrarily small outside finite sets, Proposition 2.2.9 can be formulated for t 1 (S) as follows.

The aZgebra t 1 (s) admits a bounded right approximate identity with bound M if and onZy if given £ > 0 and any finite number of eZements s 1 ,s 2 , ••• ,sn ins, there exists a e: t 1 (s) such that 2.3 PROPOSITION.

i

= 1,2, ••• ,n.

Towards our main result, (Theorem 2.7), we prove some lemmas. 139

LetS be an inve~se semigroup and e 1 , ••• ,ek Then the~e emsts ~ € t 1 (E 5 ) such that 2.4 LEMMA.

11~11 ~ Proof.

2k-l - 1

and

~·ej

= ej

fo~

€

E5 be given.

j "' 1, ••• ,k.

Recalling that E5 is a commutative semigroup an easy induction

argument on k with

yields the result. 2.5 DEFINITION.

Let S be an inverse semigroup and k a positive integer.

say E5 satisfies condition (Dk) if given u 1 , ••• ,~+l and integers i,j such that 1 < i < j

~

k+l,

eu.1

and

€

E5 we can find e

€

We E5

u •• J

If s is an inve~se semigroup~ then t 1 (E 5 ) admits a bounded approximate identity if and onLy if E5 satisfies condition (Dk) fo~ some positive intege~ k. 2.6 LEMMA.

Proof.

Suppose E5 satifies condition (Dk) for some positive integer k and

let {u1 , ••• ,un} be any finite subset of E5 •

Then condition (Dk) implies

the existence of e 1 , ••• ,~ € E5 such that for each i € {l, ••• ,n} there exists j € {l, ••• ,k} with e.u. u 1•• Then choosing ~ € t 1 (E 5 ) as in J 1 Lemma 2.4 we get e.u ... e.*~. J

1

J

l, •.. ,n.

ak*e:ll: J 1

1

Now recalling Proposition 2.3 (and the commutativity of E5 ) we conclude that t 1 (E 5 ) has a· bounded approximate identity with bound 2k-1. Conversely suppose that t 1 (E 5 ) admits a bounded approximate identity with bound M (say). Let e 1 , ••• ,ek+l € E5 be given, where k is an integer such that k > M. Proposition 2.3 says that we can find a € t 1 (E 5 ) such that

llall

140

< M

and

lle.-a*e·ll 1 1

<

1

k+l

for

i .. 1, ••• ,k+l.

With obvious notation we write

u

a = I A r r r

where ur A. :• {r 1

1 -

£

E5 and Ar e.u 1

r

I r£A.1

£

C, for all r.

For each i

{l, ••• ,k+l} let

£

• e.} and hence note that 1

IAr I -< It <

lie.1 - a*e.ll 1

So

If the sets A1 , ..• ,~+l were pairwise disjoint we would get k+l

M ~I 1Ar1 ~ I r i=l

+

which contradicts our choice of k. Hence Ai n Aj + for some i,j with 1 < i < j ~ k + 1. Thus E5 satisfies condition (Dk). 2. 7 THEOREM.

Let S be an inverse semigroup.

Then the following items

are equivalent: (i) R.l(S)

admits a bounded approximate identity;

(ii) R.l(S)

admits a bounded left approximate identity;

(iii) R.l(Es>

admits a bounded left approximate identity;

(iv) Es

satisfies condition (Dk) for some positive integer k.

Proof.

That (i) implies (ii) is trivial. Recalling that E8 (and hence R. 1 (E 8 )) is commutative, the equivalence of (iii) and (iv) is established in

Lemma 2.6. We now prove the equivalence of (ii) and (iii). Firstly, suppose R. 1 (S) admits a bounded left approximate identity with 141

bound M.

Let T : S

ES be given by T(s) := s s.

+

commutative we have, for all s

:=

T(se)

(se)"(se)

€

S and e

Then, since ES is

Es•

€

= e"s"se = e(s"s)e = (s"s)e

= T(s)e.

Hence T extends to a norm-decreasing linear map (which we also denote by) T : 1 1 (S)

+ 11 (Es)

Let e 1 , ••• ,en

such that

ES and£ > 0 be given, there exists a

€

llall < M and Then T(a)

1 1 (S) such that

l •... ,n.

II T(a) II ~ M and

1 1 (Es),

€

for i

lie.1

€

lie.1 - T(a)*e.ll .. IIT(e.) 1 1 - T(a*e.) 1 II _< lle1. - a*e1.11

<

£ •

Hence 1 1 (ES) has a bounded left approximate identity. Secondly, suppose 1 1 (ES) admits a bounded left approximate identity with bound M (say). there exists a

II all Thus for i

II iii

€

< M

Let s 1 , ••• ,s € Sand£ 1 n 1 (ES) such that and

II s.1s:1 - a*s.1 s.1 II

= 1, ... , n; - a*s.ll 1

=

as s.1

= s.s:s., 1 1 1

>

<

0 be given.

£

for i

Since sisi

€

Es•

1, ... , n.

we have that

II<~a*s.s.) 1 1 1 1 * ii.ll 1

-< lls.s: 1 1 - a*s.s.ll 1 1 Hence 1 1 (s)· admits a bounded left approximate identity. Thus (ii) and (iii) are equivalent. Since 1 1 (Es) is commutative we have (also shown) that (i) and (iii) are equivalent. From the definition of a Brandt semigroup S, (see item 4.4(b)(i)), we note that ef

~

{e,f} for distinct non-zero elements e,f

€

ES.

So, if Es is

infinite, ES does not have property (Dk) for positive integers k. Theorem 2.7 implies 142

Hence

If s is a Brandt semigroup with Es infinite~ then t 1 (s) does not have a bounded Zeft approximate identity. 2.8 COROLLARY.

The proof of the following result is contained in that of Theorem 2.7.

If s is an inverse semigroup~ then t 1 (s) admits a right approximate identity of bound 1 if and onZy if E8 is directed upwards. 2.9 PROPOSITION.

The results of this section are taken from the paper of Duncan and Namioka [28].

Theorem 2.7 and Proposition 2.9 are also related to the

results of Charles Lahr [69]. 3. A FACTORIZABLE BANACH ALGEBRA WITHOUT AN APPROXIMATE IDENTITY We recall that a Banach algebra A is said to be factorizable if every element a £ A can be written as a = be for some b.c £ A.

It is well known that a

Banach algebra with a bounded approximate identity is factorizable - recall For some time it was not clear whether. conversely.

Theorem 2.2.11.

factorization implies the existence of a bounded approximate identity. Paschke [78] showed that this converse does not hold.

In this section.

following Leinert [72]. we construct a commutative semigroup S such that t 1 (S) is factorizable but does not have an (even unbounded) approximate identity. Let S be the semigroup (with pointwise addition) of all real sequences x := {x } with x 1 n

n

> 0 for all but finitely many n £ 1N.

Consider any

f £ t (S). We can find sequences an := {an.k}k£1N and {An} such that £ s (n £ 1N) and A £ c. a n n 00

f

r.

n=l

AnX{a }. n

For each k £ 1N.

~

:=

let~ :=

em;n(...) 1

Then h :=

{~}

{an. k : 1

<

n

<

k

and

a

n.

k

>

0}

and

~ cf>.

if

~

if

~ = cf>.

£ S.

Now consider the sequence gn :=an-h.

The kth entry of gn is

143

·..• ~ {:: -

By definition of ~· we have Thus g £ 5, for all n £ m. n

~ ~ ~

Next we note that, that gn

a

for all but a finite number of k

an-h implies f • X{h} *

r

AnX{g }i so f n

n=l factorizes.

£ ~.

Hence 11 (5) is factorizable.

For every g £ 5 and f £ 11 (5) we have g not contained in the support of x{g} * f (by our definition of elements of 5) and so. llx{g}

*f

- x{g} II

~ 1.

Hence 11 (5) has no approximate unit. The reader may also note that 11 (5) is a subalgebra of a commutative group algebra and so 11 (5) is a semisimple algebra.

4. AMENABILITY OF THE ALGEBRA 11 (5) Let 5 be any discrete topological semigroup. For s £ 5, f M £ 1m(5 x 5)* we define s.f, f.s £ 1m(5 x 5) and s.M, M.s s.f(t,t') := f(st,t'), s.M(f) := M(s.f)

and

£ £

1m(5 x 5) and 1m(5 x 5)* by

f.s(t,t') := f(t,t's) M.s(f) :• M(f.s).

Let w : 11 (5 x 5) + 11 (5) be given by w(s,t) := st for all s,t ~ £ 1m(5) we define w*~ £ 1m(5 x 5) and w**M £ 1m(5)* by w*~(T)

for all T

£

:=

~(w(T))

and

w**M(~)

£

5.

For

:= M(w*~)

11 (5 x 5).

element M of 1m(5 x 5)* is called a virtual diagonal for 11 (5) if, for all s £ 5, we have

An

s.M • M.s

and

s.

11 (5) is said to be amenable if there exists a virtual diagonal for 11 (5). 144

The semigroup 5 is said to be amenabl-e if there exists a left and right invariant mean on m(S) = 1~(5). (Note that existence of a left invariant mean and a right invariant on 1~(5) implies the existence of a left and right invariant mean on 1~(5)- see e.g. Day [25].) Let e denote the usual canonical projection of 11 (5 x S) into 1~(5 x 5)*. An appPO~imate diagonaL for 11 (5) is a bounded net (ma) in 11 (5 x 5) such that, for all s € S, lim(s.e(ma ) - e(ma ).s) • 0

and

lim w(ma )*s .. s. a

a

We then have

The aLgebm 1 1 (5) has an approzimate diagonaL if and onLy if it has a vizttuaL diagonaL. 4.1 LEMMA.

Proof. Suppose 11 (5) has an approximate diagonal (denoted by) (ma). Let M be a weak*-cluster point of e(ma) in 1~(5 x S)*. We may assume that Then, for all fin 1~(5 x S), we have that

s.M(f) :• M(s.f)

lim e(ma) (s.f) a

• lim s.e(m )(f) a

a

lim e(ma).s(f) a a

= M(f.s)

• M.s(f)

and (w**M) s (+) :• w**loi(+ s ) • M(w*+s> • lim e(ma )(w*+) s a

'"' lim w*+ s (ma ) a

145

e<&><~>

for all

... (S).

Hence (w**M) s = s (on identifying S(s) with s).

~ £ R.

Thus M is a virtual diagonal for R.1 (s). By arguing in a related fashion the reader should be able to handle the converse case.

Let S be an inve~se semigroup with an identity element 1. Then the following items are equivalent:

4.2 PROPOSITION.

(i) The~e exists M in R.'"'(s x S)* such that M(l) • 1 (ii)

Proof.

and s.M

= M.s

fo~

all

s

£

S;

s is amenable. Suppose (i) holds.

We define m in R.'"'(S)* by

m(f) :• M(f) where f(s,t) :• f(s), for all f in R.'"'(s) and s,t in S. S, we have that

TI.en, for all x in

x.M(f) .. M.x
.. m(f) • Also m(l) = M(l) • 1. Thus m is a left invariant mean on R.'"'(s). Similarly, there exists a right invariant mean on R.'"'(s). So Sis amenable. Conversely, suppose (ii) holds. Let m be an invariant mean on R.'"'(s) and define M £ R.'"'(s x S)* by 146

M(f)

:=Is

Clearly M(l)

f(x",x)dm(x)

= m(l) = 1.

s.M(f) := M(s.f)

for all fin

t~(S

x S).

For each s in S and f in t~(S x S) we have

·=I . s =

Is

s.f(x",x) dm(x) f(sx· ,x) dm(x)

• Is

f(s(xs)",xs) dm(x)

=Is

f(ss"x",xs) dm(x)

• fs

f((xss")", (xss")s) dm(x)

fs = fs

f(x",xs) dm(x) f.s(x" ,x) dm(x)

• M(f.s) • M.s(f). Thus s.M

= M.s.

4.3 REMARKS. (a) One can easily note (from the proof of Proposition 4.2) that S is amenable if t 1 (s) is amenable, for any discrete semigroup S. (b) In the case of a discrete amenable group G with an invariant mean m, we have t 1 (G) amenable; in fact the virtual diagonal M can be explicitly defined by M(f) :• m(x

+

f(x,x- 1 )) for all f in i~(G x G).

[see proof of Proposition 4.2.] We will now discuss the amenability of t 1 (s) for a special class of inverse semigroups - namely Brandt semigroups. Our reference for the algebraic results on Brandt semigroups is the book of Clifford and Preston

~~147

(a) A principal series of a semigroup S is a chain

4.4 DEFINITIONS.

of ideals Si of S, beginning with s 1 := S and ending with Sm+l := that there is no ideal of s between si and si+l; (b) By a

B~t

for i

~.

such

l, ••• ,m.

semigroup we mean a semigroup S with a zero element 0

and satisfying the following axioms: (i) To each non-zero element x of S there corresponds unique elements e,f and x' in S such that ex

x,

xf

=x

and

x'x

= f.

(ii) If e and f are non-zero idempotents of S, then eSf; {0}. We recall that a Brandt

semigroup S over a group G with index set I

consists of elementary I x I - matrices over Gu{O} and a zero matrix 9 - see e.g. [20, page 102].

Writing S

= {(g) 1l ..

: g

£

G; i,j

£

I} u {9} ,

where (g) .. is the matrix with (k,t)- entry equal tog if (k,t) 1J

=

(i,j) and

0 if (k,t) ; (i,j), we get

f
=

L eh

k,

if

j

if

j ; k.

With the help of item 4.1, the reader sh0uld be able to solve the following exercises.

(See also Johnson [64] for more ge.1eral results.)

4.5 EXERCISES (i) Let T be an ideal of S and t 1 (S) be amenable.

If H is a semigroup

such that t 1 (H) and t 1 (S)/t 1 (T) are isomorphic, we have the algebra t 1 (H) amenable. 1

1

(ii) If H and S are semigroups such that t (H) and t (S) are amenable, then t 1 (H x S) is amenable.

148

(iii) Let J be a Brandt semigroup with finite index set I over a trivial group.

Writing J

M :=

I iEI

{e •• : i,j

c

<<e11 ..

e . • ) - (e

x

1 1

o

u {8}, we have that

E I}

1J

X

0

e . . ) - (e.. 1 1 11 0

is a virtual diagonal for t 1 (J), where i

E

I

e)) + (n+l){e

X

e)

is fixed and n :c

III.

Let S be a ~t semig~p oveP a gPOup F with a finite Then t 1 (S) is cunenabl.e if and onl.y if G is cunenabl.e.

4.6 THEOREM.

index set I. Proof.

0

X

0

We assume the notation on components of S introduced just after

Devinition 4.4(b). Then t 1 (G) is amenable, as noted in Remark

First, let G be amenable. 4.3(b). group.

Let J be the Brandt semigroup with index set I over the trivial Then, from Exercises 4.5(iii) and (ii), we have that

hence t 1 (J

t 1 (J)

and

G) is amenable. Noting that T := {(8,g) : g E G} is an ideal of J x G one can formulate a standard argument showing that t 1 (s) is isometrically isomorphic with t 1 (J x G)/t 1 (T). Hence, recalling Exercise 4.5(i), we have that t 1 (S) is amenable. x

Next suppose that t 1 (S) is amenable with virtual diagonal M. Let j E J be fixed. Corresponding to each f E t~(G x G) we define ~ on S x S by

~f(s,t)

.·--

~f(gl,h) 1

(s, t) • ((g) •. , (h) •• )

if

1J

1J

otherwise.

For u E G we have ~

(u) ..• f(s,t) 1J

~

= f((u) 1J ••

s, t)

~

f((ug) .• , (h) .. ) 1J

{ -

ff(ug),

-l =

1J

if

(s,t) • ((g) •. , (h) •• ) JJ

1

otherwise.

h)

if

1

otherwise.

JJ .

(s,t) • ((g) .. , (h) .. ) JJ

JJ

u:f(s,t).

149

"'

"'

Thus (u) .•• f = --.....: u.f. Similarly f.(u) •• 1J "' JJ l~(G x G) by m(f) :• M(f). Then, for all f E l~(G

x

-

= f.u.

Now let m be defined on

G) and u E G, we have

u.m(f) := m(u.f) := M(~f>

= M((u) ..• "'f) JJ

=

M
since M is a virtaul diagonal

= M(i:u> .. m(f.u). Also m(l)

= M(l) = 1.

Recalling Proposition 4.2, we have G amenable.

LetS be an inverse semigroup withES finite. Then amenable if and only if each ma:r:imal subgroup of S is amenable. 4.7 THEOREM.

Proof.

t

1 (S) is

Since ES is finite, S has a principal series

of ideals of S. We have Sm/sm+l (i.e. Sm) a simple inverse semigroup with a finite number of idempotents and so is a completely simple inverse semigroup, i.e. a group. For i l, ••• ,m-1, we have Si/Si+l 0-simple with a finite number of idempotents and so is a completely 0-simple inverse semigroup, i.e. a Brandt semigroup [20]. Hence, recalling Exercise 4.5(i), we 1 have that t (S) is amenable if and only if t 1 (Si/Si+l> is amenable, for i = 1,2, ••• ,m. Let Gm be the group Sm/sm+l and Gi the group of the Brandt semigroup Si/Si+l for i = l, ••• ,m-1. Recalling Theorem 4.6, we thus have t 1 (Si/si+l) amenable if and only if G.1 is amenable. are precisely the maximal subgroups of S.

Up to isomorphism the groups G.1 Hence our result follows.

Next we show that the conclusion of Theorem 4.6 collapses if the index set I is infinite. 4.8 THEOREM. Let S be a Brandt semigroup with an infinite inde:r: set over an arbitrary group. Then t 1 (S) is not amenable.

150

Suppose, on the contrary, t 1 (s) is amenable. Then t 1 (S) has a virtual diagonal, by Lemma 4.1. In particular t 1 (s) has a bounded left Proof.

approximate identity.

This contradicts Corollary 2.8.

So our Theorem

follows. That a discrete group G is amenable if and only if t 1 (G) is amenable was proved by Johnson [64].

The results of this section are taken from

the paper of Duncan and Namioka [28].

151

7 Non-Archimedean Fourier theory

In this chapter we depart from the frame of reference of the previous chapters by replacing the Archimedean fields E and C with a non-Archimedean field K.

An excellent reference for such studies in Functional analysis,

which we will follow closely, is the book of A.C.M. van Rooij [102] - in fact the reader may find it useful to study this chapter with the book [102] on the same desk.

The results in this chapter are taken from the paper of

Dzinotyiweyi and van Rooij [40].

As in the case with most results in non-

Archimedean harmonic analysis, these are not results to stand on their own; accordingly, we begin with a summary of some standard results and observations. 1. PREREQUISITES 1.1 DEFINITIONS.

A non-Archimedean vaLuation on K is a

Let K be a field.

function I I : K-+ :R with the following properties: (a) IxI (b) lxl

>

0 0

if and only if

x

=0

(c) lx+yl ~ max{lxi,IYil (d) lxyl

=

lxiiYI

for all x andy inK.

(K,I I> is called a non-Archimedean vaLued fieLd.

Evidently the valuation I I induces a metric on K given by p(x,y) := lx-yl

for all

x,y

E

K.

If (K,p) is a complete metric space we say (K,I I> is a compLete non-

Archimedean vaLued fieLd.

For convenience we shall simply write K in place

of (K,I I>· Also K is assumed to be complete with nontrivial valuation in this chapter. R := {x E K : lxl ~ 1} is a subring of K and I := {x E K : lxl

<

1}

a maximal ideal of ·R. The quotient k := Rfi is called the residue cLass fieLd of K and we denote the natural homomorphism R-+ k by x-+ (x E R).

x

152

The letter p will denote the characteristic of (the residue class field) k while the group of all roots of unity of K whose orders are not divisible by p is denoted by TK. x

~

The elements x,y of TK are such that lx-yl

s

1 if

y, so the natural topology on TK is discrete. We also denote the unit element of K by 1 and the K-valued characteristic

function of a set X by Xx• A non-APahimedean norm on a vector space E over K is a

1.2 BANACH SPACES. functior.

II II

E

+ :R

with the properties:

(a)

llx II

> 0

(b)

llxll

=0

(c)

llx+yll ~max{ llxll ,

(d)

II AX II

if and only if

=0

x

IIYII }

I>-I II X II

for all x and y in E and ). in K.

If E is complete with respect to the

II II , then E is called a non-APahimedean Banach spaae (over K with the norm II II ) and the dual of E, denoted by E*, is the space

metric induced by

of all continuous linear maps E

+

K.

If, in addition, E is an algebra the norm II

II is said to be poweP

mul.tipZiaative if llxnll = llxll n for all x

E E and n E lN. For commutativ 2 E, this is equivalent to : = llxll (x € E). (See e.g. [102, page One of the important results on power multiplicativeBanachalgebras 223].) now follows.

llx 2 11

Let A be a commutative Banach aZgebPa with poweP muZtipZiaative norm. Then foP evecy a E A thePe e:r:ists a Banaah aZgebPa homomoPphism ~ of A into same aompZete vaZued fieZd L that aontains K, suah that I~(a) I = II a II • 1.3 THEOREM (Springer [102, page 224]).

1.4 DEFINITION.

A subset X of a non-Archimedean Banach space E is said to

be oPthonormaZ if for every finite sequence {x1 , ••• ,xn} in X in K we have 11>.. 1x 1 + ••• + >.nxnll

llxll

= max{l>- 1 1,····1>-nl}

=1

for all

and~ 1 ••••

,>.n}

and

x in X.

153

1.5 SPACES OF FUNCTIONS.

X+ K form a linear space 1~(X), which is a Banach

The bounded maps f space under the norm

Let X be any set.

II

II~ given by

llfll~ := sup{jf(x)

I:

x EX}.

c (X) := {f E 1~(X) : for every 0

E

>

0, the set {x EX

If(x> I ~

El

is finite} is a closed linear subspace of 1~(X). Now suppose X is endowed with a topology.

The bounded continuous

functions X+ K form a closed linear subspace, C(X), of 1~(X). locally compact X, let C (X) := {f E C(X) : for every 0

{x E X : lf(x)l

~

E >

o,

For a

the set

E} is compact}.

Recall that a subset E of X is said to be cZopen if it is both closed and open in X.

Now for any topological space X and distinct a,b in X

suppose f : X+ K is a continuous function such that f(a) setting

E

~

f(b).

:= llf(a)-f(b)l, we see that {x EX: lf(x)-f(a)l ~

E}

Then is a clopen

subset of X that contains a but not b- see [102, pages 22 and 25].

Hence

the continuous functions from X into K separate the points of X if and only if the clopen subsets of X separate points.

This justifies the zero-

dimensionality requirement in Blanket assumption 1.8. The following result is the natural version of the Stone-Weierstrass theorem for the non-Archimedean case.

Let X be a ZocalZy compact ze~o­ dimensionaZ topoZogicaZ space and A a cZosed subaZgeb~a of C0 (X) such that fo~ any distinct points x and y in X we have f(x) = 0 and f(y) = 1 fo~ some fin A. Then A= C0 (X). 1.6 THEOREM (Kaplansky [102, page 218]).

1. 7 TIGHT MEASURES. _ Let X be a zero-dimensional topological space and

B(X) the ring of all clopen subsets of X. ~

A tight

measu~e

on X is a map

: B(X) + K with the properties: (a)

~

is additive.

(Hence

~(~)

(b) For all A in B(X), the set

154

= 0.) {~(B)

: B c B(X) and B C A} is bounded.

(c) If A c B(X) is shrinking (i.e. the intersection of any two sets in contains a set in A) and nA = ~. then lim ~(A) = 0.

A

AeA For each tight measure

~.

A e B(X) and x e X, let

I~I(A) := sup{I~(B)I : BeB(X), B ~A} and N~(x) := inf{I~I(O): OeB(X), xeO}

The support

of~.

supp(~).

is the closure of the set {x eX: N (x) ~

>

0}.

The set, M(X), of all tight measures on X forms a non-Archimedean Banach algebra under the norm II II given by II~ II = I~ I (X) • for all ~ in M(X) • Further, the measures with compact support are norm dense in M(X) and, if X is compact, M(X) is an isometric linear copy of C(X)*.

(All these matters

are proven in detail in [102].) 1.8 TOPOLOGICAL GROUPS AND SEMIGROUPS.

Below we briefly describe the

nature of groups and semigroups we will encounter in subsequent sections. Let G be a commutative compact topological group.

We define a K-valued

character of G as a continuous homomorphism of G into {a e K

lal

= 1}.

Then all K-valued characters of G take their values in TK if G is p-free. Below, we assume that G is zero-dimensional (so as to get the characters to separate the points of G). By o(G) we denote the set of all positive integers n for which G has an open subgroup H such that G/H contains an element of order n. is p-free if p ~ o(G).

G is compatible with Kif for every n in o(G), TK

contains n elements of order n. free;

We say G

If G is compatible with K, then G is p-

further, one of the basic results in non-Archimedean harmonic analysi5

says that the K-valued characters of G separate points of G if and only if G is compatible with K; see e.g. [102, page 360]. To avoid repetition we adopt the following convention: Blanket Assumption. For the rest of this chapter S denotes a commutative aero-dimensional topological semigroup with an identity element~ 1. We say S is torsional if every compact subset of S is contained in a compact semigroup that is a union of groups.

(In the case where S is a

topological group, every compact subsemigroup of S is a compact group (by Theorem 1.2.4) and our definition of torsionality thus coincides with that 155

employed in [102] • ) If H is a compact subsemigroup of S and G a subgroup of H, then the closure of G is again a group.

So a compact subsemigroup of S that is a

union of groups is in fact a union of compact groups. Evidently if S is torsional so is every closed subsemigroup of S. If S is torsional, we say S is p-jFee if every compact subgroup of S is p-free.

Further, S is said to be compatible with K if all compact subgroups

of S are compatible with K. 2. SEMICHARACTERS

A semicha~cte~ of S is a non-zero continuous homomorphism of S into the we denote the set of all semicharacters of (discrete) semigroup TK u {0}; s by s~. 2.1 EXERCISE.

when endowed with the topology of uniform convergence on

s~

compacta· is a zero-dimensional topological semigroup under pointwise multiplication. Our main interest in this section is to determine when, for compact S, there are enough semicharacters to separate points of S.

Towards this end

we first collect together some observations on finite semigroups. 2.2.

Let S be a finite commutative semigroup that is a union of groups. Then for every x £ S, P(x) := {x,x 2 ,x3 , ••• } is a subsemigroup of a finite group and hence is a group.

Consequently P(x) contains a unique idempotent,

which we denote by, ex. If x and y are any elements of S then exey is an idempotent;

choosing

positive integers n and m such that ex = x 0 and ey = xm, we have exey = xnym

= (xy)nm.

Hence exey = exy·

Finally, we note that the relation e < e'

if

ee' • e

defines a partial ordering

<

on ES - the set of all idempotent elements of S.

2.3 EXERCISE. (i) For all f

156

£

C(S) and x

£

S we have that xf'fx

£

C(S).

(ii) If U is a clopen subset of S then X _1 e C(S), for all x e S. Ux Apart from using the following result in Theorem 2.6, it is also of independent interest.

Let U be a ctopen partition of a compact then thezoe e:r:ists a continuous homomor-phism 1r of s into a semigzooup S; finite semi~up T, such that the pa?'tition {w-1 (t) :t e T} zoefines U. 2.4 PROPOSITION. (102, page 333]

Let O(x) be the element of U containing x, for all x in S.

Proof.

s

define an equivalence relation "' on

X"'Y

if

O(x) '"' O(y)

Since S is compact,

u is

and

Ux

finite.

-1

We

by

= Uy-1

for all

UE

u.

By Exercise 2.3(ii) each equivalence class

of the relation "' is clopen and hence the classes form a finite clopen partition V of S. For all x,y and z in S, if X"'Y• then zX"'zy. finite semigroup of all maps of 1r

:

S

~

Hence taking T to be the

V into itself, we have a well-defined map

T given by

"'

w(x)(s) :• xs

"'

for all

where y denotes the element of

s,x e S;

V containing y, for each y in S. The map

1r

is clearly a homomorphism and, for all x in S,

ys = xs n {y e S : O(ys) seS

for all

= O(xs)

x e S} and U(ys)-l

= U(xs)-l

(U e U)}

-the latter is an intersection of closed sets, so every w- 1 (w(x)) is closed, and 1r is a continuous homomorphism. {1f -l ( t) : t e T} forms a refinement of U.

Further, w-l w(x)

S O(x), so

Let S be a finite semigroup that is a union of gzooups that azoe compatibte with K. Then SA sepamtes points of s. 2.5 LEMMA.

Let a and b be any distinct points of S. Proof. item 2.2 we distinguish two cases:

Assuming the notation of

157

= eb.

Firstly, suppose ea ex e}. G :• {x £ S

For convenience, let e :• ea and Since e x e y = e xy , for all x and y inS, it follows

easily that G is a group and is compatible with K.

We have a,b

there exists a (K-valued) character, 8, of G with S(a) a map y : S

~

B(b).

£

G and so

We define

TK u {O} by Js<ex)

y(x) :•

~

L

0

(Observe that, if e

~

if

e < e

if

e

-

X

f ex.

ex• then eex

e e

ex

eex

e, so ex

£

G.)

Then y(a) = B(ea) • 8(a) I 8(b) • B(eb) = y(b). Furthermore, if x and y are points of S with e -< e xy , then e < e x e y , so It follows that y is a semicharacter of S separating a e < e and e < e • -

X

y

-

and b. Next, we suppose that e a I eb. s ~ TK u {0} given by

We may assume that ea

~

eb.

Consider

the map y :

y(x) :=

Then y

£

S~,

2.6 THEOREM.

(i)

{~

if

e

if

ea

< e

a-

X

f ex •

y(a) = 1 and y(b) = 0.

Let S be compact.

sepaPates points of

s~

This completes our proof.

Then the foZ.Z.olJJing items azae equivaZent:

s;

is a union of compact groups that are compatibZe lJJith K; (in other wrds., s is. torsionaZ and compatibZe lJJith K). (ii) S

Proof.

Suppose (i) holds.

Then there exists a natural multiplicative and s~

homeomorphic embedding of S into H :• (TK u {O}) • So we may assume that S is actually a compact subsemigroup of H. Since TK u {0} is a union of finite groups, all of them compatible with K, H is a union of compact groups, G., that are also compatible with K- say H • u G1•• For every 1

158

i£1

i € I with G.~ n S nonempty, G.~ n S is a compact subsemigroup (hence a subgroup) of the group G. compatible with K. ~

(ii) follows.

Now S

Next we assume that (ii) holds and prove (i). distinct elements of S.

=

(G. n S) and so

u

~

i€1

Let a and b be any

By Proposition 2.4, there exists a finite discrete

semigroup T and a continuous homomorphism w: S

~

T such that w(a) I w(b).

By (ii), Tis a union of groups that are compatible with K. we thus have a semicharacter yin TA such that y(w(a)) yow : S

~

TK u {0}

(i) follows.

~

By Lemma 2.5

y(w(b)).

is a semicharacter in SA separating a and b.

Hence So item

(Here yow is given by yow(x) := y(w(x)) (x € S).)

We would like to extend the following result to the semigroup case.

If G is a aommutative, aompaat and zePo-dimensionaZ topoZogiaaZ group that is aompatibZe ~ith K, then the set of aZZ (K-vaZued) ahaPaatePs of G is an oPthonormaZ base in C(G). 2.7 THEOREM (Schikhof [102, page 361]).

Recalling that for a finite set X, aaPd(X) stands for the number of elements in X, we first prove the following Lemma.

Let S be finite, a union of groups and aompatibZe ~th K. Then card(SA) = card(S) and SA is a ZineaPZy independent system in Ks. (Note that KS = i~(S) for finite S.) 2.8 LEMMA.

Proof.

By Lemma 2.5, SA separates S.

linear hull of SA is Ks.

So Theorem 1.6 guarantees that the

= card(S),

Since dim(KS)

it is sufficient to show that card(SA)

~

to complete our proof

card(S).

Recalling the notation of item 2.2, for each e € ES, let e}

and for all e' € Es n y

-1

({1})} •

Since S is a union of groups, one easily notes that {Ge}e€ES is a partition of S into groups and u card(S)

= r eeES

card(Ge)

E He e€ S

and

= SA;

card(SA) <

so

r

card(He).

(*)

e€Es 159

By Theorem 2.7, we have card (G ) • card(GA) and so (*) suggests that the e

e

result will follow if we can show that card(H) < card(GA) 1 for all e £ ES. e

-

e

The latter will follow if we can prove that; for all e £ Es• the map y

yiGe of He into

+

c;

is injective.

Let e £ ES be fixed and suppose that aiG

=

~~G

e

for some a,y £ He. e

Taking x to be any fixed element of S, we now show that a(x) = y(x).

then ee:x.

e eX e = ee X

If

e, so ex £ Ge and consequently

a(x) = a(e)a(x) = a(ex) - y(ex) - y(e)y(x) - y(x). If, on the other hand, e < e • -

X

-1 n then ext a ({1}), a(ex) ~ 1, and so a<ex) - o. But then ex= X for some n £ :N, so a(x) = 0. Similarly y(x) • 0. Hence a(x) = y(x). This completes our proof. Now we can prove the promised extension of Theorem 2.7.

Let s be compact~ a union of groups and ccmpatibZe LJi.th K. Then SA is an orothonormaZ base of C(S). 2.9 THEOREM.

Theorem 2.6 says that SA separates S and so, by Kaplansky's Theorem Proof. (see 1. 6) • the linear span of SA is dense in C(S). So it remains to show that SA is orthonormal. To this end, let y1 , ••• ,yn £SA be distinct and

A1 , ••• ,A n £ K be such that max{ lA.1. I that

I

n l:

i=l

A~y.(x)l "'1 1. 1.

Consider w : S

+

:

= l, ••• ,n}

i

= 1.

We need to show

for some x £ S.

(TK u {O})n given by (x

£

S) •

Evidently w(S) is a finite semigroup. (Recall that for A £ K with IAI ~ 1 the symbol A depicts the element of k corresponding to A - see item 1.1) 160

We now consider the maps Bi B.(w(x)) :~ y.(x) 1 1

(x

w(S) €

+

k given by

S and

i

€

{l, ••• ,n}).

Each B. is a k-valued semicharacter of w(S) and B. ~ B. for i ~ j. At least 1 1 J one ~. is non-zero, so Lemma 2.8 applied to w(S) and k in place of S and K, 1 n respectively, implies that E ~.B. ~ So there exists an x in S such i=l 1 1 that

o.

n E

A.1 B·1 (w(x)) =

i=l

I

- consequently

n

E

i=l

L:Y. (x) 1 1

n A.y. (x)

E

1

i=l

1

n

E A.y. (x) I = 1. i=l 1 1

3. THE NON-ARCHIMEDEAN MEASURE ALGEBRA M(S) Let H be any zero-dimensional topological semigroup. Archimedean) Banach algebra with respect to the norm multiplication * given by

v*~(f) for all

:=

Now for~ ~(y)

=-

II f(xy)dv(x)d~(y) II f(xy)d~(y)dv(x)

M(H) and f

v,~ €

€

C(H).

€

M(S), we define~

I

Then M(H) is a (nonII II and convolution

y

d~

(y

€

€

t~(SA) by

SA)

and recall that the map ~ + ~ is called the s. For all ~.v € M(S) we have (p*v)A = pv.

Fourie~Stiettjes

tl'ansform of

3.1 THEOREM. Let S be compact~ a union of g~oups and compatibte ~th K. Then the Fo~e~Stiettjes t~ansform of S is a Banach atgeb~ isomoPphism of M(S) onto t~(SA). Recalling Theorem 2.9 we have that SA is an orthonormal base of Proof. C(S), so we have a linear isometry, w, of c (SA) onto C(S) given by 0

161

wf :=

f(y)y

E

yeS A Now c (SA)* is just t(SA) and, since S is compact, M(S) can be identified 0

with C(S)*. lJ

Hence considering the adjoint, w*, of w we have that, for all

M(S) and f

E

(w*lJ)(f)

E C

0

(SA).,

lJ(wf)

A

=

E

f(y)J,I(y)

Yf'S

E f(y)J,I(y) =~(f). yeS A

Thus the Fourier-Stieltjes transform, lJ ~ ~ (lJ e M(S)), coincides with w*. Since w* is a linear isometry while (lJ*V)A

= ~~

(lJ,V e M(S)), our Theorem

follows. Our next Theorem cites a situation where the Banach algebra M(S) is of the type hypothesised in Theorem 1.3.

If S is torsional and

3. 2 THEOREM.

p-tpee~

then M(S) has power multi-

pticative no:rm. Proof.

By the remarks preceding Theorem 1.3 it is llll 11 2• Since tight measures with

Let lJ e M(S) be fixed. llll*'l' II

sufficient to show that

compact support are (norm-) dense in M(S), we may assume that supp(lJ) is compact.

By torsionality of S, supp(lJ) is contained in some compact sub-

semigroup of S.

So we may as well assume that S is compact.

Since K can be embedded into a complete algebraically closed field, we may also assume that S is compatible with K.

Recalling Theorem 3.1, we then

have

Remark.

It is interesting to note that even for finite S, if S is not a

union of groups, M(S) may not have power multiplicative norm.

For example

the three element semigroup {O,l,a} in which 1 is an identity, 0 a zero and a2 o, is such that (a-O)*(a-0) = 0 - the zero-measure. 3.3.

The reader can easily note that every compact subset T of S determines

an equivalent relation,

B~ y T

162

~.

T

on S

if and only if B

given by: y

on T.

Recalling that

s~

is endowed with the compact open topology we note that the by~·

equivalence classes UT, induced

form a clopen partition of

s~.

These

partitions UT' with T ranging through all compact subsets of S, generate a non-Archimedean uniformity on all bounded functions uniformity.

Let

UC(S~)

denote the Banach algebra of

K that are uniformly continuous relative to this

We urge the reader to note that

uC(S~) ~ C(S~)

(a)

s~ ~

s~.

and if S is a topological group then our definition of

a uniformly continuous function on S

coincides with that employed in [102].

(b) If S is torsional then for every clopen partition U, that is uniform in the above sense, there exists a compact subsemigroup T of S such that UT refines U. (c) Our notation gives the impression as if the uniformity is determined by the structure of

On the contrary, this is not the case, as shown

s~.

in Item 5.2. We wish to identify, for a large class of S, the range of the FourierStieltjes transform.

Towards this end we have the following Proposition.

3.4 PROPOSITION. Let s be toPsionaZ and s~ separate the points of s. the FoUPieP-Stieltjes tPansfoPm maps M(S) isometPically into UC(S~). Proof.

Let T be 3ny compact subsemigroup of S that contains 1.

show that semicharacters of T extend to semicharacters of S. end,

let~

be the restriction

As

maps~+ T~.

T~

T, Theorem 2.6 says that T is compatible with K; base of C(T), (by Theorem 2.9), containing element

ofT~,

T we must have C(T) T~

~(S~)

either Be

= span

RT(S~).

orB L span

~(S~),

so

~(S~).

Then

We first

To the latter

separate the points of T~

is an orthonormal

Thus if B is any Since

RT(S~)

by Theorem 1.6 and so Be

separates

~(S~).

Thus

= RT(S~). Now fix

~

e M(S) with compact support.

T with 1 e T and

supp(~)

c T.

We can find a compact subsemigroup

As above, T is compatible with K and so

163

lhdl

sup{ If Sdlll : S

E

y

sup
supI

II~ II

y

E

by Theorem 3.1

T~}

s~}

E

since

T~

• R (S~) T

since supp(lJ) c T

s~}

•

Evidently ll is constant on the elements of the partition UT and so ~ E UC(S~).

Since measures with compact supports are dense in M(S), our

result follows. We now present our desired target by simply sharpening the conclusion of our preceding Proposition.

For the case of a topological group see e.g.

[102, Theorem 9.20].

Let S be torsional and s~ separate the points of s. Then the Fourier--StieZtjes tPansfoPm is a Banach aZgebPa isomorphism of M(S) onto 3.5 THEOREM.

UC(S~).

Proof.

In view of Proposition 3.4 it is sufficient to show that

A :• {f

E UC(S~)

end let g

: f =~for some ll

E UC(S~)

E

M(S)} is dense in UC(S~).

and£> 0 be given.

To this

Recalling Remark 3.3(b), we can

find a compact subsemigroup T of S containing 1 such that lg(S)- g(y)l ~ £

for all

s.y

E s~

with SIT- YIT.

Corresponding to each element Ia of the partition liT choose one and only one Ya

E

Ia and consider the function f f(S) • g(ya)

if and only if

s~ +

S

E

Ia•

K given by for each Ia

E

UT.

Then f is constant on elements of UT and lg(y)- f(y)l ~ £ So our result will follow if we show that f E A. As noted in the proof of Proposition 3.4 every element of an element of s~ and so we may define h E t~(T~) by

164

for ally

T~

E s~.

extends to

By Theorem 3.1, we can find v E M(T) such that h(B) =IT Bdv for all BE TA. Defining p E M(S) by p(E) := v(E n T) for all clopen E S S, we have f(y) Thus f

h(yiT)

=I YIT dv =I

p and f E A.

ydp =

~(y)

As remarked above, this completes our proof.

4. THE NON-ARCHIMEDEAN SEMIGROUP ALGEBRA M (S) n

Let H be any zero-dimensional topological semigroup. X E

For x E H we define

M(H) by i(U) := { -01

if

X E

if

X

U

~ U,

for all U E B(H).

(With the context being different we do not expect the reader to confuse this notation with that in item 1.1.) Let M!(H) := {p

E

M(H)

the map x -+ x*p of H into M(H) is norm continuous}

Mr(H) := {p n

E

M(H)

the map

X

-+ P*x of H into M(H) is norm continuous}

M (H) := M1 (H) n Mr(H). n n n For a zero-dimensional topological group G, Schikhof [102, Theorem 8.9] showed that if M1 (G) ~ {0} then G is locally compact and carries a K-valued Haar measure

~

n

.t n

j~

such that M (G) = M (G) = L(G) := {p E M(G) : p n

for

some continuous j : C -+ K} - hence the reason for our interest in sets like

First we make some remarks on integration of vector valued functions. For all v E M(H), all Banach spaces E and all bounded continuous functions f : H -+ E, in a reasonable way one can define an integral f f(x)dv(x) that is an element of E. If E and F are Banach spaces and T :E -+ F is a linear continuous map, then 1

I

T(f(x))dv(x)

= T(ff(x)dv(x)). 1

In particular noting -

that M (H) is clearly a Banach space, i f E := M (H) and f(x) := x*P n

1

n

for

some P E Mn(H), since the maps y-+ y*(i*p} = yx*p of H into M(H) are clearly continuous (x E H) f is thus a continuous function of H into M1 (H) n

165

and so we obtain an element


dv(x))(A)

v*~ =I x*~

I i•~dv(x)

of M!(H).

For any clopen ASH,

=I x*~(A)dv(x) = v*~(A),

dv(x).

Thus

v*~

€

M!(H).

More generally we have the

following result.

Fol' any zero-dimensional topological semigroup H, we have that and Mn (H) al'e closed ideals of M(H).

4.1 THEOREM. M1 (H) n Proof.

It is sufficient to prove the result for M1 (H).

closed.

Let ~

v*~

Evidently M1 (H) is

n

E

M!(H) and v

E

H we have

E

M(H) be fixed.

n

Then as observed above

Mn1 (H).

£

For all x,y

and so

~*v

!

EMn (H).

This completes the proof.

Taking SA := {y E SA: ~(y) ~ 0 for some ~ E M (S)} and M (S)A to be the n n n set of all non-zero multiplicative linear functional& on M (S), we have the n

following non-Archimedean analogue of Theorem 2.5.6. SA with the compact-open topology is homeomol'phic toM (S)A n n

4.2 THEOREM.

with the Gel'fand topology. Proof.

Let hE Mn(S)A be fixed and choose

Consider yh : S yh(x) :•

~

~ ~

Mn(S) such that

h(~) ~

0.

K given by

h(x*~> h(~)

Then, as in-the proof of Theorem 2.5.3, one can easily show that yh E S~ and that each y E SAn arises from a unique h E M (S) with yh = y; thus the n map h ~ yh of Mn(S)A into SA is both one-to-one and onto. n We now show that the map h ~ yh is a homeomorphism in a similar but simpler way to the proof of Theorem 2.5.6. Suppose ha yh

~

a

166

~

h in Mn(S)A and let T c S be compact.

yh uniformly on T.

We can choose

~

E Mn (S) with

We wish to show that h(~) =

1 and may

assume that, for all a, we have jha(~)- h(~)j

<

1.

Then

and

so jha(~)j

1. Now the elements ha (~)-lh a of Mn (S)* form a norm bounded net that converges in the weak*-topology; so it converges uniformly on the compact subsets of Mn (S). As {x*~ : x E T} is a compact subset, we have that uniformly for x ~

Thus yh

E

T.

yh uniformly on T.

a

Next, suppose ya ~ y in s~. For each v E M (S) and £ > 0 we can choose n n p E M(S) such that T := supp(p) is compact and llv-p II < £. For large enough a, we have Ya • y on T, so f yadp = f y dp and (with ya = yh and a y .. yh)

Hence h

a

~

h in M (S)~. n

4.3 THEOREM. Let s be to~sional and compatible with K. Then the canonical map of Mn (S) into t•(SA) is an isomet~ic algeb~ homomoFphism. Proof.

First we note that the map ~ ~ ~ of M (S) into t~(S~) has the n

properties (v,~ E

M (S)). n

Since S is compatible with K, it is p-free and so M(S) has power multiplicative norm, by Theorem 3.2.

So for every ~ E M (S) a result of n Springer.(see Theorem 1.3) implies the existence of a multiplicative linear

map

~

: M(S)

~

IH~> I ..

L, where L is some complete field containing K, such that II~ II

and

II~ II

..

1.

167

Setting y(x) := ~(x) we have a bounded homomorphism y : S ~ L. y(x)~(~)

Now

= ~(x)~(~) = ~(x*~)

(x £ S), so y is continuous. Since S is compatible with K, it follows that y(x) £ K for all x £ Sandy£ SA. Recalling the remarks preceding Theorem 4.1, for any v £ M(S), we have

~(v)~(~) = ~(v*~) = ~
=I y(x)~(~)dv(x) So ~(v) = v(y), in particular~(~)

11~11 = IH~>I = I~I ~ and so

11~11

= lliill.

=

= ~(y).

v(y)~(~). Hence

IIOII

This completes our proof.

A comparison of Theorems 4.3 and 3.5 motivates one to seek for the image of M (S) under the Fourier-Stieltjes transform. Towards this end we first n give a definition. 4.4 DEFINITION. A continuous function f : SA ~ K is said to be a:l'bitrur>i'ty sma'tt outside equicontinuous sets if given £ > 0 we can find an equicontinuous set A C SA such that If I ~ £ on SA \ A; the set of all such f is denoted by EQ (SA). 0

The following lemma is also of independent interest.

4.4 LEMMA.

We have the following inclusion relations.

(ii)

Proof. equivalence relation x ~ y

if

Given ~

£

>

0, we consider an

on S defined by

y(x) = y(y)

for all

y £ SA with lf(y) I ~

£.

The equivalence classes are evidently clopen and hence the quotient semigroup R :• Sf~ is discrete. Let w : S ~ R be the quotient map and w: RA ~ SA the dual map of n. Since RA is compact (as R is discrete) and

n is 168

continuous, we have that n(RA) is a compact subset of SA.

By our

definition of ~we have lfl < f E C (SA).

E

outside the compact set n(RA) and so

0

(ii) It is sufficient to show that a characteristic function of a compact and open subset of SA is in UC(SA). Let V c SA be compact and open. Recalling the definition of compact open topology on SA, for each a E V we can find a compact Ka c S such that

(and o(a) is a neighbourhood of a).

So for some n E lN, we have

a1 , ••• ,an E V such that V = O(S1 )u ••• uo(an).

Setting T := Ka u ••• uKB 1

we

n

see that V is a union of elements of the clopen partition UT (defined in item 3.3) and so the characteristic function of V lies in UC(SA).

(i) The M (S) onto

s be

Let

4. 5 THEOREM.

to~sionat

and SA

Fourie~-Stiettjes t~nsfo~

is a Banach

atgeb~

s.

Then

isomoPphism of

EQ (SA).

n

0

(ii) If every compact subset of SA is

Stiettjes

points of

sep~te

t~sfo~

Proof. (i) Given A:= {y E SA :

~

is a Banach atgebra

E M (S) and

n IPI ~

E}.

E

then the Fou~e~­ of Mn (S) onto C0 (SA).

equicontinuous~

isomo~phism

> 0, we consider the set

For all x,y E S

llx*~- Y*~ll >sup l<x*~- Y,..~)A(Y)I =sup jy(x)- y(y>II~I -yEA

YEA

>

E

sup ly(x) - y(y)l; yEA

hence A is equicontinuous. By Lemma 4.4(i) and Theorem 3.5 we have that the map ~ + ~ is an isometri algebra homomorphism of M (S) into EQ (SA). To show that this map is onto, n 0 fix f E EQ (SA) and choose v E M(S) such that f = (which can be done, by

v

0

Theorem 3.5 and Lemma 4.4). We now show that v E Mn(S). Fix x £ S and E > 0. By our choice of f the set A:= {y E SA

lv(y)l ~ E} is equicontinuous, so

O(x) := {y E S : ly(x) - y(y) I <

II vii

-l£ for all y E A} is an open 169

neighbourhood of x.

Now, for all y

IY(X) - y(y>ll~(y) I

(*)

{

£

~ llvll -l

O(x) we have llvll =

£

£

if y

A

<.

< 1.£ = £

Recalling Theorem 3.5, for all y £ O(x), we have lli*v- y*vll =

llAII =sup l
I

sup IY(X) - y(y)l lvl y£SA < £

by(*).

•

Consequently v £ Mn(S) and item (i) is proven. Item (ii) now follows from item (i) and Lemma 4.4(i) together with the fact that now C (SA) cEQ (SA). 0

4.6 REMARKS.

-

0

Even in the case of topological groups, the requirement that

every compact subset of SA be equicontinuous employed in Theorem 4.5(ii) is not redundant.

We refer the reader to [102, page 356] for an example :

There, a torsional zero-dimensional topological group G is constructed such that GA is compact but G is not locally compact. C0 (GA) = C(GA) ~ {0}.

Thus M (G) = {0} n

~1hich

Nevertheless the class of topological semigroups S for subsets of SA are equicontinuous is large.

while

compact

Below we cite some of such

cases: (a)

Suppose S (is torsional and) has the property : for every subset 0 of ~ S ~e have that 0 is open. (This

S with 0 n T cZopen in T for aZZ compact T

property is met, for example, if s is locally compact or metrizable and Then e11ery compact compares closely with the definition of a k-space.)

subset A of SA is equicontinuous. Proof.

Let A be a compact subset of SA.

show that O(x) := {y £ S : y(y)

= y(x)

Our result will follow if we

for ally£ A} is clopen (x £ S).

To this end, let T be a compact subsemigroup of S with 1 the restriction map SA

170

+

£

T and note that

TA is continuous and TA is discrete.

Hence

A1

:~

{yiT : Y

€

A} is (compact and so) finite.

n {y y£A

O(x) n T "'

€

T : y(y) = y(x)}

so O(x) n T is clopen in T. (b) Suppose

{y n Y£Al

a

Now €

T

y(y) ,. y(x)} •

By the given property of S, O(x) is clopen.

s (is tol'siona.Z.

and) has the pl'OpePty: fol' eVef'Y x £ sand

y £ SA thel'e is a neighboul'hood U of x and an a £ S such that Ua is

PeZ.ativeZ.y compact and y(a) I 0. continuous. Proof.

Let

A~

Then compact subsets of SA aPe equi-

SA be compact and x

S be fixed.

€

We show that O(x),

defined as in case (a), is a neighbourhood of x. For every y £ A, let UY be a neighbourhood of x and ay an element of S such that Uy a y is relatively compact and y(ay ) I 0. By compactness of A, there exists finitely many points y 1 , ••• ,yn in A such that Ac

n u {B i=l

€

SA : B(a

)

I

0}.

yi

Let U :• U n••• nu and, for convenience, write a. in place of a and notE 1 Y1 Yn Yi that Uai is relatively compact, fori • l, ••• ,n. Since S is

to~sional,

there is a compact subsemigroup T of S containing

the (compact) set {l,a1 , ••• ,an} u (U{a1 , ••• ,an}). of (a), {yiT : y B1 , ••• ,Bm

€

€

A such that

for each B £A, BIT= BjiT for some j Now W := {y

As noted in the proof

A} is finite, so there exist finitely many points

€

€

{l, ••• ,m}.

U: B.(y) .. B.(x) for j = l, ••• ,m} is indeed a neighbourhood J

J

of x, so A will be equicontinuous at x if we can show that W ~ O(x). To the latter end, let y i

€

€

Wand B

€

A.

There exist j

€

{l, ••• ,m}

and

{l, ••• ,n} such that and

B(a.) I 1

o.

Since

171

S(ya.)

S.(ya.) J

1

In particular S(x)

1

=

S.(y)S.(a.) J

J1

Sj(x), whence S(y)

=

S.(y)S(a.) and so S(y)

= S(x)

J

1

andy£ O(x).

=

S.(y). J

Thus W c O(x)

and we are done. In general, the properties of S stated in (a) and (b) are distinct - in fact neither implies the other;

for counter-examples, we refer the

interested reader to [40; item 4.8]. It ~emains an open p~oblem ~hethe~ one can obtain a gene~l ch~cte~zation of those semig~ups s fo~ which compact subsets of SA ~e equicontinuous. 5. NOTES 5.1.

It would be interesting if one could discover some meaningful non-

Archimedean analogues of the spaces Ma (S) and Me (S) studied in Chapter 2. As noted above the object Mn(S) seems to be quite interesting and a lot remains to be studied - for instance, it is not clear what the "foundation" of such an object is.

We however suspect that it could be reasonable to

define the foundation of A S M(S) as the smallest closed set containing the set {x £ 5.2.

s : Nll (x)

~ 0

for some

ll

£.A}.

The structure of SA does not determine the uniformity defined in

commutative topological semig~ups S and T with identity elements and such that: SA and TA ~e identical as topological semig~oups but UC(SA) ~ UC(TA). Eee e.g. [40, item 3.3.

In fact one can find

ze~o-dimensional

Example 4.4]. 5.3.

A lot needs to be investigated on S •

For instance if S is locally

compact, to~sional and compatible with K, we conjectu~e that SA sep~ates points of S. See also [40, Section 41 for some examples on the structure of SA. 5.4.

As noted at the beginning of this chapter, the results studied here

are taken from the paper of Dzinotyiweyi and van Rooij [40].

The reader

who wishes to research more on abstract harmonic analysis of this type may find it helpful to study related results in the group case. the latter include the theses of H.W. Schikhof, 172

References for

"Non-A~chimedean ha~onic

analysis", Katholieke Universiteit, Nijmegen (1967); A.M.M. Gammers [47] and L. Duponcheel, "Non-Archimedean induced representations and related topics", Vrije Universiteit Brussel (1979).

173

Appendix A The strict topology and duality th.eorem Throughout this Appendix X denotes a C-distinguished topological space. 1. THE STRICT TOPOLOGY A.l.

A non-negative ~(x) ~

{x £ X :

hoods on X.

function~

in m(X) is called a hood if the set

£} is compact, for every

>

£

0.

Let Hd(X) be the set of all

The strict topoLogy (or equivalently, the B-topoZogy) on C(X) is

the topology generated by the seminorms plf.(f) :=

ll~fll X= sup{~(x) lf(x)

for all f in C(X)

and~

strict topology.

in Hd(X).

{p~

I:

~

£ Hd(X)}, where

x £X}

We write

c 8 (X) for C(X) endowed with the

(See e.g. [12], [61] and [103) for more details on the

strict topology.) 2. A C-ENERALIZED STONE-WEIERSTRASS THEOREM

Let A be a B-cZosed subaZgebra of C(X) z.Jhich separates points and contains, fo?' ·~u~h x in X, a fun.:!~ion nonzero at x. Then A • C(X). A.2.

THEOREM (Giles [43]).

We are particularly interested in a Corollary of this Theorem (given below). Towards the latter end we prove a lemma. A.3.

of X.

Let

LEMMA.

~

be a positive measure in M(X) and F a compact subset

Then

(i)

~(F) ~ inf{~(f)

: f £ C(X) is a positive function with Xp

~

f}

(ii) C(X) is a dense subspace of L 1 (x.~). Proof. that

(i) Let

~(U)

<

£

>

0 be given and choose an open neighbourhood U of F such

p(F) + IE.

{C } c X\ U such that n that 174

We can find an increasing sequence of compact sets ~

CD

( u C ) • n=l n

~(X\

U) •

So we can choose n 0 £

~

such

Since F and Cn are disjoint compact sets and X is C-distinguished, a standaJ compactness argument shows that there is a real-valued function fn such that

Hence, for n

~

n0

,

€

C(X)

we get

~(F) ~ f fn(x)d~(x)

•

f (x)d~(x) + U n

f

< ~(F)

f

X\U

f

n

(x)d~(x)

+ £.

This completes our proof for item (i). (ii) Let A be any Borel subset of X. We first show that XA is a cluster point of C(X) in L 1 (X,~). To this end, let £ > 0 be given. There is a compact set F and an open set U such that F

~A~

U and

~(U\F) <

!£.

Now

let the sequence of compact sets {Cn} and of functions {fn} be chosen as in (i) with respect to F and u. We then get llxA-fnlll := a

JlxA-fnl(x)d~(x)

fU\F

lxA-f

~ 2~ (U\ F)

+

n

=

J0 1xA-fnl(x)d~(x)

l(x)d~(x)

~ ((X\U)\

for

+

fx\UixA-fnl(x)d~(x)

+

f(X\U)\ en f n (x)d~(x)

>

n •

Cn) n

-

0

It is now evident from the first paragraph of the proof of Hewitt and Ross [59, Theorem 12.10] that item (ii) holds. A.4. COROLLARY. Let A be a subatgeb~ of C(X) such that A sep~tes points and foP each x in X some membep of A does not vanish at x. Then A is dense in L 1 (x.~) foP every positive measupe ~in M(X).

175

Proof. Let~ be a given positive measure in M(X). Suppose f £ L1 (X,~) and £ > 0 are given. By item (ii) of our Lemma, we can find g £ C(X) with

flf-gl(x)d~(x)

1£.

<

There is an increasing sequence {Kn} of compact subsets such that K) • n

function

II~ II

1jJ :=

·

I

n•l pljl(g-h) <

Setting K0 :=

+ and

anxK,K 1 is a hood. n n-

1£

for some

h

an :=

~(K

n\Kn-1 ) we have that the

Then, by Theorem A.2,

A.

£

Hence

flf-hl(x)d~(x) ~ Jlf-gl(x)d~(x) <

1£

+

+

J

;

n=l

Kn\Kn- l

J1g-hl(x)d~(x)

lg-hl(x)d~(x)

3. A GENERALIZATION OF THE RIESZ-REPRESENTATION THEOREM The following Theorem is due to Giles [43] for the case where X is completely regular. The present form was noted by Hirschfeld [61]. A.S. THEOREM. Fo~ each ~ in M(X) let h be the jUnctional on C(X) given by h (f) :a Jf(x)d~(x) (f £ C(X)). T~n the map~ ~ h~ defines a ~ line~ isomorphism fr'om M(X) onto c6 (X)* - (the dual of c 6 (:~)). A. 6. (i)

COROLLARY. II~ II

..

We have that

sup{ l~(f) I

:

f

£

C(X) and

llfll X~

1} fo~

the no:rom II II , is a Banach space. c 6(X)* is a closed subspace of C(X)*.) (ii) M(X), uncle~

aU ~ in M(X).

Equivalently

Item (i) follows trivially from Lemma A.3(ii). Now M(X) is clearly Proof. a (complex) normed linear space, so to prove item (ii) it remains to show that M(X) is complete. To this end, suppose {~n} is a Cauchy sequence in M(X). Then item (i) implies that {~n} is a Cauchy sequence in the Banach space C(X)* and so converges to some ~ in C(X)*. 176

We now show that p is strictly continuous. choose n0 such that

IIPn -pll C(X)*

< £.

Let

Since pn

0

there is a hood p~(f) <

1

~

£

>

0 be given and

is strictly continuous,

0

such that if (f

then

E

C(X)).

Hence, for such f, we have IP(f)l

<

IPn (f)l +

£

< £(

llfllx + 1).

0

Thus p is strictly continuous.

By Theorem A.4 we must have p in M(X) and

our result follows.

177

Appendix B Weakly compact sets

1, WEAKLY COMPACT SUBSETS OF A BANACH SPACE

Let c be a subset of a Banach space Then the following items are equivalent:

B.l.

A.

THEOREM (Eberlein-Smulian).

(i) C is Pelatively weakly compact; (ii) each sequence in C has a weakly convergent subsequence in

A (i.e.

C

is sequentially weakly compact). The reader can find the proof of the preceding Theorem in e.g. Dunford and Schwartz [29, V.6.1].

A proof for the following Theorem can be found

in Glicksberg [ 46] • B.2. THEOREM (Krein-Smulian). If Cis a weakly compact subset of a Banach space A then the closUPe of the convex hull of c is weakly compact. The next Theorem is proved by Kelley and Namioka [66, page 160].

The weak topology foP a pPOduct of lineaP topological spaces is the product of the weak topologies foP the cooPdinate spaces. B.3.

THEOREM.

2. WEA.~Y COMPACT SUBSETS OF M(X) The equivalence of the first two items of our next Theorem is due to Grothendieck !55, Theorem 2] and that of the last two items can be deduced from Stein [99, pages 139 and 140]. (In fact the whole Theorem is essentially due to Grothendieck [55].) B. 4.

THEOREM.

subspace of M{X). (i)

A is PeZativeZy weakly compact;

(ii) given

178

Let X be a locally compact topological space and A a Then the folLowing items aPe equivalent:

E >

0 we have

A satisfying the conditions

(a) thePe exists some compact K ~ X such that I~I(X\K) <£foP att ~ E

coPresponding to any compact 0 of K such that

(b)

1~1 (0\K) <£foP att ~

(iii) thePe is a sequence

that

~n(E)

B.S.

A,

= 0 (n

E

:IN)

NOTATIONS.

{~n}

we have Let

A1

X

thepe is an open neighbouPhood

A;

E

A such

in ~(E)

and

c

K

= 0

A2

that~ (~ E

foP att BoPet E c X such A).

be normed linear spaces.

A linear map

T of A1 into A2 is said to be weakty (or no~) compact if it carries every bounded subset of A1 into a weakly (or norm) compact subset of A2• The following Theorem is a special form of the Dunford-Pettis property see e.g. Day [26, page 108] and Dunford and Schwartz [29] for the general cases.

B.6.

THEOREM.

Tl : M(X)

-+-

Let X,Y and

M(Y) and T2 : M(Y)

-+-

composite tineaP map T2oT1 : M(X)

z be compact topotogicat spaces and M(Z) weakty compact UneaP maps. -+-

Then the

M(Z) is norm compact.

3. WEAKLY COMPACT SUBSETS OF C(X)

B. 7.

a subspace of C(X). (i)

Let X be a topotogicat space and A Then the fottowing items aPe equivatent:

THEOREM (Grothendieck [54]).

A is Petativety weakty compact;

(ii) if {fn} and {xm} aPe sequences in A and X, Pespectivety~ then we have that the ctosuPes of the sets {fn (xm) : n < m} and {f n (xm) ; n > m} aPe not disjoint~ (oP equivatentty wheneveP the doubte timits a := lim lim f (x ) n m n m and b :=lim lim f n (x) m exist~ we must have a= b). m

n

B.S. NOTATION. Let X and Y be topological spaces and P(X) := { v E M(X) : v is positive and II vII • 1}. Corresponding to each function F in C(X x Y) we define the map F' : P(X) x P(Y) -+- C by

F'(~,v)

:=If F(x,y)dv(x)d~(y)

(v

E

P(X)

and~

E

P(Y)).

179

For all x £X and v £ P(X) the functions F(x,.) and F'(v,.) defined on Y and P(Y), respectively, are given by F(x, .)(y) := F(x,y)

and

F' (v, .)(lJ) := F' (v,'IJ)

for ally£ Y and ll £ P(Y). Similarly one defines the functions F(.,y) and F'(.,'IJ) on X and P(X), respectively, for ally£ Y and ll £ P(Y). Evidently F(x,.) £ C(Y) and F(.,y) £ C(X) for all x £X andy£ Y. generally we have

B.9. above.

More

Let X and Y be topoZogicaZ spaces and F and F' as defined

LEMMA.

Then

(i) The map x

+

F'(x,ll) of x into cis continuous ('ll £ P(Y)).

(ii) If the set {F(x,.) : x £X} is reZativeZy ~akZy compact~ then the map x + +(F(x,.)) of X into Cis continuous <+ £ C(Y)*). Proof.

(i) Given £ > 0, let K be a compact set such that 'IJ(X\K)

a :c IIFII xxy· that

4£a where For each x £ X we can find a neighbourhood O(x) of x such

IF(x,y) - F(t,y)l < }£

<

for all t £ O(x) andy£ K.

Hence, for all t £ O(x), we have IF'(x,'IJ)-F'(t,ll)l = !JI 5._ J IF(x,y) - F(t,y) ldll(y) + J

K

IF(x,y) -F(t,y) ld'IJ(y) X\K

~ }'IJ(K)£ + 2 IIFII XxY 'IJ(X\K) <

£.

(ii) Let (xa) be a net converging to a point x in X. Since the net (F(xa,.)) is relatively weakly compact, it has a weak cluster point fin C(Y). For each y £ Y we have F(xa ,.)(y) := F(xa ,y) Thus F(x,.) 180

=f

+

F(x,y)

and so

f(y) = F(x,y).

is the only weak cluster point of (F(xa,.)) and so our result

follows.

l'HEOREM.

B.lo. C(X

X

Let X and Y be topoZogicaZ spaces and F a function in RecaZZing the notation in B.8, we have the foZZowing items

Y).

equival-ent: (i) The sets {F(x,.) : x £X} and {F(.,y)

compact in C(Y) and C(X),

Y} aPe reZativeZy weakl-y

y £

~spectiveZy;

(ii) the sets {F'(v,.) : v £ P(X)} and {F'(.,~) : ~ £ P(Y)} aPe reZativeZy weakl-y compact in m(P(Y)) and m(P(X)), ~spectiveZy. Proof.

That (ii) implies (i) is trivial.

v £ P(X).

£ C(X)

(Indeed F(v,.) £ C(Y), by Lemma B.9(i).)

{F(x,.) : x £X} weakly compact. a

v(f) for all f £ C(K).

lj>(F(A ,.)) a

=f +

But F(A a ,y)

=f

a

f !jl(F(x, .) )dv(x)

f !jl(F(x,.))dv(x) = !jl(F(v,.))

of the convex hull of

{x :

x £ K} such that

So, by Lemma B.9(ii), we have

lj>(F(x, .))dA (x)

F(x,y)dA a (x)

61 ,

Fix v £ P(X) with K := supp(v) compact.

We can find a set (A ) in the convex hull of +

Similarly define

£ P(Y).

for~

By Theorem B.2 we have the closure, call if

Aa (f)

For

Let F(v,.) £ C(Y) be the function given by F(v,y) := F'(v,y)

for all y£Y. F(.,~)

Now suppose (i) holds.

since F' is linear (!jl £ C(Y)*)

F(x,y)dv(x) (!jl £ C(Y)*).

regularity of measures we then have F(v,.) £

= F(v,y)

(y

Thus F(v,.) £

"'61 ,

£

Y).

"'61 •

Hence By the inner

for all v in P(X).

Hence {F(v,.) : v £ P(X)} is a relatively weakly compact subset of C(Y). Recalling the symmetric criterion for weak compactness (see Theorem B.7), this is equivalent to saying {F'(.,y) : y £ Y} is a relatively weakly

6

compact subset of m(P(X)). So the closure, say 2 , of the convex hull of {F'(.,y) : y £ Y} is weakly compact. By arguing in a similar manner to the

.

precedLng paragraph, we get that F'(.,~) £ "' 62 for all~£ P(Y). Consequentl {F'(.,~) : ~ £ P(Y)} and (by Grothendieck's Theorem B.7), {F'(v,.) :v£ P(X)' are realtively weakly compact subsets of m(P(X)) and m(P(Y)), respectively.

181

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47(1973), 147-159. 70. R. Larsen: Transformation groups and measures with separable orbits.

Monatsh.

Math. 73(1969), 222-225. 71. A.T. Lau: Invariant means on subsemigroups of locally compact groups.

Rocky

Mountain J. Math. 3(1973), 77-81. 72. M. Leinert: A commutative Banach algebra which factorizes but has no approximate units.

~c.

Amer. Math. Soc. 55(1976), 345.

73. T.S. Liu, A. van Rooij and J.K. Wang: Group representations in Banach spaces: Orbits and almost periodicity.

Studies and essays presented to Yu-Why Chen on his sixtieth birthday (1970)' 243-254.

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74. I. Namioka:

Studia Math. 29(1967),

On certain actions of semigroups on L-space. 63-77. 75. A. Olubummo:

Finitely additive measures on the non-negative integers.

Math.

Scandinavica 24(1969), 186-194. 76. A. Olubummo: A decomposition theorem for finitely additive measures on a discrete

NigePian J. Science 4(1970), 101-110.

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Topological

semig~ups.

Math. Centre Tracts, Amsterdam (1970).

78. W.L. Paschke: A factorable Banach algebra without bounded approximate unit.

Pac.

J. Math. 46(1973), 249-251. 79. A.L.T. Paterson: Invariant measure semigroups.

P~c.

Lond. Math. Soc. 35(1977), 313-

332. 80. A.L.T. Paterson: Amenability and locally compact semigroups.

Math. Scand. 42(1978),

271-288. 81. J.S. Pym: Convolution algebras are not usually Arens-Regular.

Quapt. J. Math

Oxford (2) 25(1974), 235-240. 82. J.S. Pym: The convolution of functionals on spaces of bounded functions.

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Lond. Math. Soc. 15(1965), 84-104. 83. J.S. Pym: Weakly separately continuous measure algebras. (1968), 207-219. 84. J.S. Pym·and H.L. Vasudema: An algebra of finitely additive measures.

Math. Annalen 175

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86. Roger Rigelhof: Invariant measures on locally compact semigroups.

Proc.

Ame~.

Math.

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90.

91.

92.

93.

94.

95.

96.

97.

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190

Studia Math.

Index of symbols

Meaning

Page 98

a given space of functions on a semigroup S a compactification of S arising from

A8

non-zero multiplicative linear functions on -1

-1

98

A

38

the set {A1A2 , A1 A2 , A1A2 }

2

non-negative elements of the space A

4

non-positive elements of the space A

4

B(X,Y)

set of all functions of X into Y

3

Ba

certain operators of Ma(S) into LUC(S)

BA(S)

bicontinuous points of A(S)

C(X,Y)

continuous functions in B(X,Y)

3

CK(X,Y)

functions in B(X,Y) continuous on compacta

3

C(X)

bounded functions in C(X,C) or C(X,K)

3

C (X)

functions in C(X) vanishing at infinite

3

C

functions in C(X) vanishing outside compacta

3

0

oo

(X)

66 100

C(X) endowed with the strict topology

174

Arens operations

101

density function arising from a set of measures

4

(Dk)

an ordering type of condition on E 8

140

EMR.(S)

left emaciated measures in M(S)

120

EQO(SA)

functions SA ~ K arbitrarily small outside equi-continuous sets

168

set of all idempotent elements in S

1

191

6~isolated idempotent& in a semigroup S

51

the closure of a set E

4

vof, fov

convolutions of a function f and measure v

5

Fa(S), Fe(S)

foundations of Ma(S), Me(S) (respectively)

28

Hd(X)

set of all hoods on X

IM(WUC(S))

set of all invariant means on WUC(S)

K

non-Archimedean valued field K

152

k

residue class field of K

152

R.l(S)

discrete semigroup algebra of S

1

R. (S,w)

174

weighted discrete semigroup algebra of S

76

19 133

translation maps of ll defined on E c S

13

translation maps of ll defined on S

13

L1 (G)

the group algebra of a locally compact group G

11

L1 (G,w)

weighted group algebra of G

L1 (X,p)

space of p-integrable functions on X

4

L(X,p)

measures in M(X) absolutely continuous to ll

4

R.

p,E

R. • r ll

r p,E

ll

LMC(S)

functions f in C(S) with aof all a

LUC(S) (RUC(S))

E

133

E

C(S) for

135

93

left (right) uniformly continuous functions in C(S)

63

left (right) weakly uniformly continuous functions in C(S)

63

LIM

left invariant mean

71

M(X)

bounded Radon measures or tight measures on X

4

M*

the continuous dual of a normed linear space M

5

M!(s), M~(S), Ma(S)

spaces of absolutely continuous measures on S

11

M1 (S), Mr(S), Me(S)

spaces of equi-regular measures on S

26

spaces of measures norm continuous on translation

21

LWUC(S) (RWUC(S))

e

e

R.

r

Mn(S), Mn(S), Mn(S)

192

M (S)A n

non-zero multiplicative linear functionals on M (S) n

Mil. (S) eq-a '

M"eq-a (S)}

166

sets of equi-absolutely continuous measures 117

Meq-a(S)

on S

MJI. (S,H) eq MJI.(S) q

left equi-quasi-invariant measures

35

left quasi-invariant measures

35

m(X)

space of bounded complex-valued functions on X

3

Ill I

measure arising from total variation of ll

4

Ill I*

inner measure arising from llll

4

ll

the completion of a measure ll

4

lliB

restriction of a measure ll to set B

4

ll

the Fourier transformation lJ on M(S)

lll*ll2

the convolution of two measures lll and ll 2

N (x)

inf{llli(O) : 0 an open neighbourhood of x}

ll

II II II II X

+

~ defined

40 7

155

norm (e.g. total variation norm on M(X))

4

supremum norm on space of functions on X

3

o(G)

{n £ 1N : G/H has an element of order n, for some open subgroup H}

155

pljl

a pseudo-norm on m(X) arising from

174

P(H) -LIM

a mean left invariant with respect to P(H)

s

given (topological) semigroup [in most casesJ

sl

smallest dense ideal of a stip S

46

sr

the set {xES : for all f £ CK(X,Y), f X E C(S,Y) for any topological spaceY}

17

continuous semicharacters of S

38

1jl £

m(X)

non-Archimedean continuous semicharacters that are not Mn(S)-negligible supp(lJ)

72 1

16~

4

the support of a measure ll 193

TLIM

topological left invariant mean roots of unity not.divisible by characteristic of k

72

153

uniformly continuous functions on S

63

non-Archimedean uniformly continuous functions on SA

163

non-Archimedean uniformity induced by a compact set T S S

163

WAP{S)

weakly almost periodic functions on S

63

WUC{S)

weakly uniformly continuous functions on S

63

X

point mass at x for x image of x € K in k.

€

S, or the natural

XE

characteristic function of a set E

x0 -clo{F)

countable closure of a set F

194

4 4

53

Subject index

absolutely continuous measure 11

foundation semigroup

28

action, anti-action (of a semigroup) 57

Fourier transform 40, 161

almost periodic function

Grothendieck's theorem for (a

amenability

70

72, 144

A-measurable

4

A-negligible

4

compact set of) measures

178

Grothendieck's theorem for (a

weakl~

compact set of) continuous

approximate diagonal 145

functions

approximate identity 23

group 1

Arens operations 93, 101

hood

bicontinuous points

idempotent 1

100

179

174

Brandt semigtoup 147

identity element 1

a-topology

(left) invariant mean 71

174

cancellation element

weakl~

inverse semigroup 139

1

C-distinguished space 4

Krein Smulian theorem 178

C-distinguished topological semigroup 5

k 0 -space 3

centre of a set 1

k-space

characteristic function 4

(left) locally quasi-invariance 127

3

clopen set 154

L-ideal 8

Cohen's factorization theorem 25

locally relatively neo-compact 2

compact open topology 40

L-subalgebra 8

compatibility with K

mean 71

155, 156

convolution measure algebra

8

multiplicative linear functional 38

countable closure (of a set)

53

non-Archimedean Banach space 153

o-isolated idempotent

non-Archimedean norm 153

51

discrete semigroup algebra Eberlein-Smulian theorem

132 178

non-Archimedean valuation 152 non-Archimedean valued field 152

(left) emaciated set 120

p-free 155

equi-absolutely continuous measure 117

power multiplicative norm 153

equi-quasi-invariant measure 35

principal series 148

equi-regular measure 26

quasi-invariant measure 35

foundation of a set of measures 8

radical of an algebra 42 195

regular multiplication 101 relatively neo-compact 2 residue class field 152 Riesz-representation theorem 176 semicharacter 38 semigroup

1

semi-simple algebra

42

semitopological semigroup 2 separable left orbit 57 solid set 5 stip 45 Stone-Cech compactification

as

144

Stone-Weierstrass theorem 174 strict topology 174 subgroup 2 subsemigroup 1 support of a function 4 support of a measure

4

tight measure 154 topological (left) invariant mean 72 topological group 2 topological semigroup 2 torsional semigroup 155 translation invariant 8 uniformly continuous function 63 virtual diagonal 144 weak (or strong) convergence [to topological invarianceJ

76

weak approximate identity 23 weakly almost periodic function 63 weakly uniformly continuous function 63 weight function 132 zero element 1

196