Almost periodic functions on groupoids

Almost periodic functions on groupoids

FARIDBEHROUZI(*)

ABSTRACT- In this paper we generalize the notion of almost periodic functions on groups to the corresponding notion for groupoids. We prove a number of theo- rems about almost periodic functions in this general setting. We show that the set of almost periodic functions on a groupoidG,AP(G), is a C*-subalgebra of

`¹(G). We investigate some topological properties of the maximal ideal space of AP(G),b(G), and we obtain a continuous partial operation onbG. Also, we study almost periodic functions on groupoids defined by an equivalence relation on a setXand obtain a compactification ofX.

MATHEMATICSSUBJECTCLASSIFICATION(2010). 22A22.

KEYWORDS. Groupoid, almost periodic function.

1. Introduction

For a locally compact spaceX, every unital C*-subalgebra AofC_b(X) induces a compactification ofX. In fact the maximal ideal space of Ais a compactification ofX, see [4]. HereC_b(X) is the algebra of bounded con- tinuous complex-valued functions on locally compact spaceX. It is inter- esting to know that ifXhas further algebraic structures, which compact- ifications ofXpreserve these algebraic structures. For example, ifSis a discrete semigroup andAˆ`¹(S)ˆC_b(S), then the algebraAinducesbS, the Stone-CÏech compactification ofS, which is a compact right topological semigroup, see [7]. Another example is the algebra of almost periodic

(*) Indirizzo dell'A.: Department of Mathematics, Alzahra University, Vanak, Tehran, Iran p.code:1993891176.

School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746, Tehran, Iran.

E-mail: behrouzif@yahoo.com

This research was in part supported by a grant from IPM (No. 90470015).

functions on a groupG. Recall that a bounded functionf on a groupGis said to be almost periodic if the set of all left translationsfLxf :x2Ggoff has compact closure in`¹(G), whereLxfis defined byLxf(y)ˆf(xy). This algebra induces the Bohr compactification ofGwhich is denoted bybG. It is proved thatbG is a compact group and there exists a homomorphism f:G!bGsuch thatf(G) is dense inbG. In fact (bG;f) is a group com- pactification forG. This compactification is the biggest compactification, in the sense that if (K;c) is another group compactification forG, then there exists a continuous homomorphismT:bG!K such that Tfˆc. For more details, see [1].

The notion of almost periodic functions on (R;‡) was created by Harald Bohr in [3]. The notion is a generalization of purely periodic functions. This notion was extended to arbitrary groups by Bochner and J. Von Neumman in [2] and thereafter this theory was extended to semigroups. For further notes on almost periodic functions see; e.g. [5] and [1].

This note serves two purposes. First, we want to generalize the defi- nition of almost periodic functions on groups to groupoids. The main dif- ficulty in carrying out this construction is that the product operation on a groupoid is partially defined and the definition of translation of a function is not trivially generalized. In Section 3, we define the translation of a function on groupoids and then we define almost periodic functions on groupoids. The second aim of this paper is to obtain a compactification for a groupoidGby showing that the set of all almost periodic functions onG, AP(G), is a C*-subalgebra of`¹(G). This fact is shown in Corollary 3.5.

Also, we investigate the topological and algebraic properties of the max- imal ideal spaceAP(G). In Section 4, we examine the results in the previous section for groupoids arising from equivalence relations on a setXand in this setting we get a compactification forX.

2. Preliminaries and notations

Let X be an arbitrary set. We denote by `<sup>1</sup>(X) the set of bounded complex-valued functions on X. So, `¹(X) with the pointwise operations and the norm

kfk₁ˆsup

x2Xjf(x)j

is a unital commutative C*-algebra. Let X be a Hausdorff topological space. ThenC_b(X) stands for the C*-algebra of bounded continuous com-

plex-valued functions on X. We will replace Cb(X) by C(X) when X is compact.

In this paper, we are interested in compact subsets of`¹(X). The fol- lowing lemma which is an easy consequence of Proposition 13.1 in [11], will be frequently applied in the sequel.

LEMMA2.1. LetCbe a subset of`<sup>1</sup>(X). Then the closure ofCin`¹(X), C, is compact if and only if for anye>0there exist subsets X₁;X₂; ;X_n of X such that XˆX₁[X₂[ [X_n with the property that if x;y2X_j, for some1jn, thenjf(x) f(y)j<efor all f 2 C.

Here are some elementary definitions in groupoid literature. For more details, we refer the reader to [8], [9] and [10].

A groupoid is a set G endowed with a product map (x;y)7!

xy:G⁽²⁾!GwhereG⁽²⁾is a subset ofGGcalled the set of composable pairs, and an inverse mapx7!x ¹:G!Gsuch that the following prop- erties hold:

(i) If (x;y)2G⁽²⁾ and (y;z)2G⁽²⁾, then (xy;z)2G⁽²⁾;(x;yz)2G⁽²⁾ and (xy)zˆx(yz).

(ii) (x ¹) ¹ˆx, for allx2G.

(iii) For allx2G, (x;x ¹)2G⁽²⁾and if (z;x)2G⁽²⁾, thenz(xx ¹)ˆz.

(iv) For allx2G, (x ¹;x)2G⁽²⁾and if (x;y)2G⁽²⁾, then (x ¹x)yˆy.

The maps r and d on G, defined by the formulae r(x)ˆxx ¹ and d(x)ˆx ¹x, are called the range map and the source (domain) map, respectively. A pair (x;y) is composable if and only ifr(y)ˆd(x). The unit space G⁰ is defined by G⁰ˆd(G)ˆr(G). Its elements are units in the sense thatxd(x)ˆr(x)xˆx. Units will usually be denoted byu;v;wwhile arbitrary elements will be denoted byx;y;z. The fibres of the range and the source maps are denoted by G^uˆr ¹(fug) and Gvˆd ¹(fvg) re- spectively. Also foru;v2G⁰,G^v_uˆGu\G^v.

3. Generalization of almost periodic functions

In this section we define almost periodic functions on a groupoid G.

Since in the usual definition of almost periodic functions on groups the translations of functions play basic role, first we define translations of functions on groupoids.

DEFINITION 3.1. Suppose that u2G⁰, x2Gu and y2G^u. For f 2`¹(G) we denote the leftx-translation and the righty-translation off, respectively, byL^u_xf andR^u_yf and we define

L^u_xf :G^u!C; R^u_yf :G_u!C;

L^u_xf(t)ˆf(xt); R^u_yf(t)ˆf(ty):

ClearlyL^u_xf 2`<sup>1</sup>(G<sup>u</sup>),R<sup>u</sup><sub>y</sub>f 2`¹(G_u).

THEOREM3.2. Let G be a groupoid and let u2G⁰. Then the following properties of f 2`¹(G)are equivalent:

(i) fL^u_xf :x2G_ugis compact in`¹(G^u).

(ii) fR^u_xf :x2G^ugis compact in`¹(G_u).

PROOF. (i)) (ii). Suppose that fL^u_xf :x2Gug is compact in `¹(G^u) and lete>0. By Lemma 2.1, there exist subsetsX1;X2; ;XnofG^usuch that ift;t⁰2X_j for some 1jn, then for anyx2Gu,

jL^u_xf(t) L^u_xf(t⁰)j<e:

Now take a fixed t_i in the setX_i( iˆ1;2; ;n); hence, for anyt2G^u, there is somet_j2G^usuch thattandt_jare both inX_j, accordingly, for any x2G_u, we have

jf(xt) f(xtj)j ˆ jL^u_xf(t) L^u_xf(tj)j<e;

and hence

kR^u_t f R^u_tjfk₁<e:

This argument shows that the set fR^u_xf :x2G^ug is totally bounded in

`<sup>1</sup>(G<sub>u</sub>). Therefore,fR<sup>u</sup><sub>x</sub>f :x2G<sup>u</sup>gis compact in`¹(G_u).

(ii))(i). The proof of this is the same as (i))(ii). p DEFINITION3.3. LetGbe a groupoid. A functionf is said to bealmost periodicfunction if for anyu2G⁰,fL^u_gf :g2G_ugis compact in`¹(G^u).

The set of almost periodic functions onGis denoted byAP(G).

THEOREM3.4. Let G be a groupoid and let f;g2AP(G). Then (i) f‡g; fg;f 2AP(G)and AP(G)contains the constant functions.

(ii) Ifff_ng¹_nˆ1is a sequence in AP(G)such that f_n!f in`¹(G), then f 2AP(G).

PROOF. (i) Let e>0 and let u2G⁰. By Lemma 2.1, there exist subsets X₁⁰;X⁰₂ ;X_n⁰ and X₁⁰⁰;X⁰⁰₂; ;X_m⁰⁰ of G^u such that G^u ˆ

Sⁿ

iˆ1X_i⁰ˆ Sⁿ

jˆ1X_j⁰⁰and ift;t⁰2Xi, then for anyx2Gu, jL^u_xf(t) L^u_xf(t⁰)j<e;

and ift;t⁰2X_j⁰⁰, then for anyx2Gu,

jL^u_xg(t) L^u_xg(t⁰)j<e:

Set X_i;jˆX_i⁰\X_j⁰⁰ (iˆ1;2; ;nand jˆ1;2; ;m ). Ift;t⁰2X_i;j, then for anyx2Gu;

jL^u_x(f ‡g)(t) L^u_x(f‡g)(t⁰)j jL^u_xf(t) L^u_xf(t⁰)j

‡ jL^u_xg(t) L^u_xg(t⁰)j

<e‡eˆ2e:

Also, we have:

jL^u_x(fg)(t) L^u_x(fg)(t⁰)j ˆ jf(xt)g(xt) f(xt⁰)g(xt⁰)j jf(xt)kL^u_xg(t) L^u_xg(xt⁰)j

‡ jg(xt⁰)kL^u_xf(t) L^u_xf(xt⁰)j kfk₁e‡ kgk₁e:

Therefore, by Lemma 2.1, the sets fL^u_x(f‡g):x2Gug and fL^u_x(fg):x2Gugare compact in`¹(G^u).

(ii) Let e>0. Pick N2N such that for any t2G, jf_N(t) f(t)j<e.

Since f_N2AP(G), there exist subsets X₁;X₂; ;X_n of G^u such that G^u ˆ Sⁿ

iˆ1X_iand ift;t⁰2X_i, then

jL^u_xf_N(t) L^u_xf_N(t⁰)j<e:

Ift;t⁰2X_i, then we have:

jL^u_xf(t) L^u_xf(t⁰)j jL^u_xf(t) L^u_xfN(t)j ‡ jL^u_xfN(t) L^u_xfN(t⁰)j

‡ jL^u_xf_N(t⁰) L^u_xf(t⁰)j

ˆ jf(xt) f_N(xt)j ‡ jL^u_xf_N(t) L^u_xf_N(t⁰)j

‡ jfN(xt⁰) f(xt⁰)j

<e‡e‡eˆ3e:

p

COROLLARY 3.5. Let G be a groupoid. Then AP(G) is a unital C*- subalgebra of`¹(G).

Let bG be the maximal ideal space of AP(G). Then bG is a compact Hausdorff topological space under the Gelfand topology and by the Gel- fand-Naimark theorem,AP(G) is isometrically *-isomorphic toC(bG). We defineW:G!bGbyW(x)(f)ˆf(x). It is easy to see thatWis well-defined andW(G) is dense inbG.

We can easily extend the functionsL^u_xfandR^u_xf to whole ofG. For any x2G, we define

Lxf :G!C; Rxf :G!C;

Lxf(t)ˆ f(xt) t2G^d(x) 0 t2=G^d(x) (

Rxf(t)ˆ f(tx) t2G_r(x) 0 t2=G_r(x): (

Therefore, the functionsLxf andRxf are in`¹(G) andLxfj_Gd(x) ˆL^d(x)_x f andRxfj_Gr(x) ˆR^r(x)_x f.

THEOREM3.6. Let G be a groupoid and let f 2AP(G). Then for any x2G, the functionsLxf andRxf are in AP(G).

PROOF. Letx02G. A straightforward verification shows that L^u_x(L_x0f)ˆ L^u_x0xf x2G^d(x0)

0 x2=G^d(x0): (

Therefore,

fL^u_x(Lx0f):x2Gug fL^u_x0xf :x2G^d(xu ⁰⁾g [ f0g fL^u_xf :x2Gug [ f0g:

Since fL^u_xf :x2G_ug [ f0g is compact so is fL^u_x(L_x0f):x2G_ug.

Similarly, it is proved thatR_xf is inAP(G). p Leth2bGand letf 2AP(G) . We define two functionsTh;f :G!C andSh;f :G!Cby

T_h;f(x)ˆh(Lxf) ; S_h;f(x)ˆh(Rxf):

By Theorem 3.6, the functions T_h;f andS_h;f are well-defined and are in

`¹(G).

THEOREM3.7. Let G be a groupoid and f 2AP(G).

(i) Ifh2W(G^u)for some u2G⁰, then T_h;f 2AP(G).

(ii) Ifh2W(G_u)for some u2G⁰, then S_h;f 2AP(G).

PROOF. (i) Let fxag_a2I be a net in G^u such that W(xa)!h. Hence, T_W(xa);fj_GuˆR^u_xaf ˆ R_xafj_GuandR_xafj_Guis pointwise convergent toT_h;fj_Gu. Since the closure offR^u_xf :x2G^ugunder the norm topology is compact, the closure of fR^u_xf :x2G^ug under the norm topology is the same as the closure of fR^u_xf :x2G^ug under the pointwise topology. Therefore, T_h;fj_Gu2 fR^u_xf :x2G^ugand henceR_xafj_Guis norm convergent toT_h;fj_Guin

`¹(G_u). On the other hand, ift2=G_u, thenT_h;f(t)ˆ R_xaf(t)ˆ0, thusR_xaf is norm convergent toT_h;f. By Theorem 3.6 and Corollary 3.5,T_h;f 2AP(G).

(ii) The proof is similar to (i). p

COROLLARY 3.8. Let G be a groupoid. If u;v2G⁰ and u6ˆv, then W(G^u)\W(G^v)ˆ ;.

PROOF. Leth2W(G^u)\W(G^v). Choosef 2AP(G) withh(f)ˆ1. Since v2=G^u,T_h;f(v)ˆ0. Letfxag_Ibe a net inG^vsuch thatW(xa)!h. So

T_h;f(v)ˆlim

a T_W(xa);f(v)

ˆlim

a f(xa)

ˆlim

a W(xa)(f)ˆh(f)ˆ1:

This is a contradiction. p

COROLLARY 3.9. Let G be a groupoid and let u2G⁰. Suppose that f 2AP(G).

(i) If h2W(G^u) and h_a2W(G^u) with h_a!h, then T_ha;f !T_h;f in AP(G).

(ii) If h2W(Gu) and h_a2W(Gu) with h_a!h, then Sh_a;f !Sh;f in AP(G).

PROOF. (i) By Theorem 3.7, T_ha;f and T_h;f are in AP(G), hence Th_a;fj_Gu;Th;fj_Gu are in fR^u_x:x2G^ug. Since Th_a;f is pointwise convergent toT_h;f in`<sup>1</sup>(G) andfR<sup>u</sup><sub>x</sub>:x2G<sup>u</sup>gis compact, thusT<sub>ha;f</sub>j<sub>Gu</sub>converges to T<sub>h;f</sub>j<sub>Gu</sub> in the norm topology of`¹(G_u). On the other hand, T_ha;fj_(GnGu) ˆ

Th;fj_(GnGu) ˆ0. Therefore,Tha;f converges toTh;f in the norm topology of AP(G).

(ii) The proof is similar to (i). p

THEOREM3.10. Let G be a groupoid andh;u2bG. Letfx_ag_Iandfy_bg_J be nets in G_uand G^u, respectively, such thatW(x_a)!handW(y_b)!u. Then

lima lim

b W(xayb)ˆlim

b lim

a W(xayb):

PROOF. Letf 2AP(G). We have:

lima lim

b W(x_ay_b)(f)ˆlim

a lim

b f(x_ay_b)

ˆlim

a lim

b W(yb)(Lxaf)

ˆlim

a u(Lxaf)ˆlim

a W(xa)(T_u;f)

ˆh(T_u;f):

By Corollary 3.9 (i),T_W(yb);f !T_u;f inAP(G). Therefore, h(T_u;f)ˆlim

b h(T_W(yb);f)

ˆlim

b lim

a W(x_a)(T_W(yb);f)

ˆlim

b lim

a f(xayb)

ˆlim

b lim

a W(x_ay_b)(f):

p COROLLARY 3.11. Let G be a groupoid. If h2W(Gu) and u2W(G^u), then

h(T_u;f)ˆu(S_h;f):

PROOF. We have:

h(T_u;f)ˆlim

b lim

a W(x_ay_b)(f)

ˆlim

b lim

a W(xa)(Rybf)

ˆlim

b h(R_ybf)ˆlim

b W(y_b)(S_h;f)

ˆu(Sh;f):

p

For a groupoidG, set (bG)⁽²⁾₀ ˆ [

u2G⁰

W(G_u)W(G^u); (bG)₍₀₎ˆ [

u;v2G⁰

W(G^v_u);

and

(bG)⁽²⁾₁ ˆ [

u;v;w2G⁰

W(G^v_u)W(G_w^u):

We define an operation ``'' on (bG)⁽²⁾₀ byhu(f)ˆh(Tu;f), where (h;u)2 (bG)⁽²⁾₀ .

THEOREM 3.12. Let G be a groupoid and u2G⁰. Then the map (h;u)7!hu:W(G_u)W(G^u)!bG is continuous.

PROOF. Let e>0, f 2AP(G) and (h₀;u0)2W(Gu)W(G^u). Let V ˆ fg2bG:jg(f) h₀u₀(f)j<eg. By Corollary 3.9 (i), the map u7!T_u;f :W(G^u)! AP(G) is norm continuous. Therefore, there exists an open set U₁ in bG containing u₀ such that for any u2U₁, kTu;f Tu0;fk<e=2. Let U2ˆ fg2bG:jg(Tu0;f) h₀(Tu0;f)j<e=2g. For (h;u)2U2U1,

jhu(f) h₀u0(f)j ˆ jh(Tu;f) h₀(Tu0;f)j

khk:kT_u;f T_u0;fk ‡ jh(T_u0;f) h₀(T_u0;f)j e:

LetK be a set and letK⁽²⁾be a subset ofKK. Iff 2K, we set K_f ˆ fg2K:(g;f)2K⁽²⁾g; K^f ˆ fg2K:(f;g)2K⁽²⁾g:

DEFINITION 3.13. A Jordan semigroupoid is a triple (K;K⁽²⁾;), such thatKis a set,K⁽²⁾is a subset ofKKandis an operation fromK⁽²⁾toK.

Let (L;L⁽²⁾;) and (L⁰;L⁰⁽²⁾;⁰) be Jordan semigroupoids. A map T:L!L⁰is called homomorphism if (f;g)2L⁽²⁾, then (T(f);T(g))2L⁰⁽²⁾ andT(f g)ˆT(f)⁰T(g).

THEOREM3.14. Let G be a groupoid and let(K;K⁽²⁾; ?)be a Jordan semigroupoid such that K is compact Hausdorff topological space.

Suppose that

(i) For any f 2K, the map g7!gf :Kf !K and the map g7!fg:K^f !K are continuous,

(ii) c:G!K is a homomorphism such that for any u2G⁰, c(Gu)c(G^u)K⁽²⁾.

Then there exists a unique continuous homomorphism F:bG!K such thatFWˆc.

PROOF. LetF2C(K). We want to show thatFc2AP(G). To do this, consider the map (t;s)7!F(c(st)):c(Gu)c(G^u)!C. By assumption, this map is separately continuous and hence the mapx7!LsF:c(Gu)! C(c(G^u)) is norm continuous; see [1, Lemma B.3]. By continuity, the set fLsF:s2c(Gu)g is a compact subset of C(c(G^u)) and hence the set fLc(x)F:x2Gughas compact closure inC(c(G^u)). So, by applying Lemma 2.1, we conclude that the set fLx(Fc):x2Gughas compact closure in

`¹(G^u), as required. Then it is easy to verify that the map T:C(K)! AP(G) defined byT(F)ˆFc is a *-homomorphism. IfFˆTj_bG, then F:bG!Kis continuous andFWˆc. For the last equality, suppose that x2G. For anyF2C(K), we have:

F(F(W(x)))ˆW(x)(T(F))ˆFc(x)ˆF(c(x)):

Therefore, FWˆc. Now, assume that (h;u)2W(G_u)W(G^u) for some u2G⁰. Letfx_ag_Iandfy_bg_Jbe nets inG_u andG^u, respectively, such that W(x_a)!handW(y_b)!u. Then

F(h)ˆlim

a F(W(x_a))ˆlim

a c(x_a):

SoF(h)2c(G_u). Similarly,F(u)2c(G^u), and hence (F(h);F(u))2c(G_u) c(G^u)K⁽²⁾. Also, we have:

F(h)?F(u)ˆlim

a lim

b F(W(xa))?F(W(y_b))

ˆlim

a lim

b c(x_a)?c(y_b)

ˆlim

a lim

b c(xayb)

ˆlim

a lim

b F(W(x_ay_b))ˆF(hu):

p THEOREM3.15. Suppose that G is a groupoid and(h;u)2(bG)⁽²⁾₀ .

(i) Let u2(bG)₍₀₎. Then hu2W(Gv) if and only if u2W(G^u_v) for some u2G⁰.

(ii) Let h2(bG)₍₀₎. Then hu2W(G^w) if and only if h2W(G^w_u) for some u2G⁰.

PROOF. (i) Suppose thatu2W(G^u_v) andfxag_a2Jandfyag_a2Jare nets in GuandG_v^u, respectively, such thatW(xa)!handW(ya)!u. Thenxaya2Gv

and hencehuˆlim

a W(x_a)W(y_a)2W(G_v). Conversely, lethu2W(G_v) and u2W(G^u_v0). This implies thathu2W(Gv)\W(Gv⁰). By Corollary 3.5,vˆv⁰.

(ii) The proof is similar to (i). p

LetGbe a groupoid. Iff 2AP(G), definebf :G !Cbybf(x)ˆf(x ¹). It is easy to see thatbf 2AP(G). We extend the inverse operation `` 1'' onGto an involution on bG. If h2bG, define h ¹(f)ˆh(bf). Since fgb ˆbfbg, h ¹2bGand the map h7!h ¹ is continuous. Also, if h2f(G_v^u), then by Theorem 3.12,h ¹hˆvandhh ¹ˆu.

DEFINITION3.16. LetLbe a set andL⁽²⁾be a subset ofLL. An op- eration :L⁽²⁾ !L is called to be associative if f;g;h2L are such that either

(i) (f;g)2L⁽²⁾and (g;h)2L⁽²⁾, or (ii) (f;g)2L⁽²⁾and (fg;h)2L⁽²⁾, or (iii) (g;h)2L⁽²⁾and (f;gh)2L⁽²⁾

then all (f;g);(g;h);(f g;h) and (f;gh) lie inL⁽²⁾, and (fg)hˆf (gh):

THEOREM3.17. Let G be a groupoid. Then (i) the operation:(bG)⁽²⁾₁ !bG is associative.

(ii) ifh2W(Gu), thenhuˆh.

(iii) ifh2(bG)₍₀₎, then(h;h ¹)2(bG)⁽²⁾₁ and(h;h ¹)2G⁽⁰⁾.

4. Equivalence relations as groupoids

LetXbe any set and letRbe an equivalence relation onX. Set R⁽²⁾ˆ f((x;y);(y;z)):(x;y);(y;z)2 Rg;

define (x;y)(y;z)ˆ(x;z) and define (x;y) ¹ˆ(y;x). Then it is easy to check that R becomes a groupoid and it is evident that R⁽⁰⁾ is the diagonal Dˆ f(x;x):x2Xg. Also, forx2X,

G^(x;x)ˆ fxg [x]; G_(x;x)ˆ[x] fxg

where [x] is used to designate the equivalence class determined byx. Two extreme cases are to be singled out. IfR ˆXX, thenRis called the trivial groupoid onX, while ifR ˆD, thenRis called the co-trivial groupoid onX.

Let Rbe an equivalence relation on a setX. So, f 2`¹(R) is almost periodic if and only if for any x2X, the setff_s:s2[x]g is compact in

`¹([x]), wheref_s:[x]!Cis defined byf_s(t)ˆf(s;t).

DEFINITION4.1. LetXbe any set. A functionf 2`¹(XX) is called piecewise cylindrical if there exists a partitionX₁;X₂; ;XnforXsuch that for any 1kn;s2Xandt;t⁰2X_kimplyf(s;t)ˆf(s;t⁰). We denote the set of piecewise cylindrical functions onXXbyPC(X).

THEOREM4.2. Let X be any set and letRbe an equivalence relation on the set X. Then for any x2X, PC([x])is dense in AP(R)j_[x][x].

PROOF. First, we show that PC([x]) is contained in AP(R)j_[x][x]. Suppose that F2PC([x]). Thus there exists a partition X1;X2; ;Xn

for [x] such that for any s2[x] and for any 1kn the identity F(s;t)ˆF(s;t⁰) holds for all t;t⁰2X_k. If we define f⁰:R !C by

f⁰(s;t)ˆ F(s;t) (s;t)2[x][x]

0 (s;t)2= [x][x];

(

then f⁰j_[x][x]ˆF. We claim thatf⁰2AP(R). Let e>0 and y2X. Con- sider subsets X₁;X₂; ;X_n;X_n‡1ˆXn[x]. By assumption, if t;t⁰2X_k, then for alls2[y],jf⁰(s;t) f⁰(s;t⁰)j ˆ0<e. Therefore, by Lemma 2.1 the setff⁰s :s2[y]g is compact in`¹([y]). Thusf⁰ is almost periodic. Now, suppose thatf 2AP(R) and suppose thate>0. By Lemma 2.1, there exists a partitionX₁;X₂; ;X_nfor [x] such that ift;t⁰2X_k for some 1kn, then for alls2[x],jf(s;t) f(s;t⁰)j<e. Lett₁2X₁;t₂2X₂; ;t_n2X_nbe fixed points. Define F:[x][x]!C byF0(s;t)ˆf(s;tk) if t2Xk. Triv- ially,F02PC([x]) and ift2X_kfor some 1kn, then

jF₀(s;t) f(s;t)j ˆ jf(s;t_k) f(s;t)j<e:

p LetRbe the co-trivial groupoidD. Trivially, everyf 2`<sup>1</sup>(R) is almost periodic. So,AP(R)ˆ`¹(R) and hencebR ˆbD. More generally, ifRis an equivalence relation on a setXsuch that every equivalence class ofRis finite, thenAP(R)ˆ`¹(R) and hencebR ˆbD.

THEOREM4.3. LetRbe the trivial groupoid on a set X. Then bR ˆ bXbX.

PROOF. We prove the theorem in the following steps.

Step 1. Iffx_ag_a2J;fy_ag_a2J;fx⁰_ag_a2Jandfy⁰_ag_a2Jare nets inXsuch that lima xaˆlim

a x⁰_a and lim

a ya ˆlim

a y⁰_a (the limits are taken in the bX), then lima W(x_a;y_a)ˆlim

a W(x⁰_a;y⁰_a).

To show that lim

a W(xa;ya)ˆlim

a W(x⁰_a;y⁰_a) inbR, it is enough to show that for everyf 2PC(X),

lima W(xa;ya)(f)ˆlim

a W(x⁰_a;y⁰_a)(f):

Let X₁;X₂; ;Xn be a partition for X such that if t;t⁰2X_k for some 1kn, then for anys2X,f(s;t)ˆf(s;t⁰). By Theorem 6.5 in [6], the setsX₁;X₂; ;X_n(the closures are taken inbX) are open and disjoint sets, so, we can assume that there exists 1jn such that for any a2J, ya;y⁰_a2X_j. Suppose thatxa!uinbXandtis a fixed element inX_j. Define g(s)ˆf(s;t). Thusg2`¹(X) and we have:

lima W(xa;ya)(f)ˆlim

a f(xa;ya)

ˆlim

a f(xa;t)ˆlim

a g(xa)

ˆu(g):

Similarly, lim

a W(x⁰_a;y⁰_a)(f)ˆu(g).

Step 2. If fW(xa;ya)g_a2J and fW(x⁰_a;y⁰_a)g_a2J are nets in bR such that lima W(xa;ya)ˆlim

a W(x⁰_a;y⁰_a), then lim

a xaˆlim

a x⁰_aand lim

a yaˆlim

a y⁰_a. Letg2`¹(X). By Lemma 4.2 the mapf defined byf(s;t)ˆg(s) is an almost periodic function onR. So

lima g(xa)ˆlim

a W(xa;ya)(f)

ˆlim

a W(x⁰_a;y⁰_a)(f)

ˆlim

a g(x⁰_a):

Thus lim

a xaˆlim

a x⁰_a. By the same proof, lim

a yaˆlim

a y⁰_a.

Leti_b :X!bXbe the injection map. Therefore,bXbXis compact semigroupoid (in fact it is groupoid as trivial groupoid on bX) and

ibib:XX!bXbX is a continuous homomorphism which sat- isfies all the conditions of Theorem 3.14. So, there exists a homo- morphism c:b(XX)!bXbX such that cWˆi_bi_b. We show that c is an isomorphism. Let u₁ and u₂ be in b(XX) such that c(u₁)ˆc(u₂). Letfx_ag_a2J;fy_ag_a2J;fx⁰_ag_a2Jandfy⁰_ag_a2Jbe nets inXsuch that lim

a W(x_a;y_a)ˆu₁ and lim

a W(x⁰_a;y⁰_a)ˆu₂. We have:

lima i_bi_b(x_a;y_a)ˆlim

a cW(x_a;y_a)

ˆc(u1)ˆc(u2)

ˆlim

a cW(x⁰_a;y⁰_a)

ˆlim

a i_bi_b(x⁰_a;y⁰_a):

Therefore, lim

a x_aˆlim

a x⁰_aand lim

a y_aˆlim

a y⁰_a. Thus u₁ˆlim

a W(x_a;y_a)ˆlim

a W(x⁰_a;y⁰_a)ˆu₂:

Now, suppose that (g₁;g₂)2bXbXand suppose thatfx_agandfy_agare nets inbXsuch thatxa!g₁ andya!g₂. By passing to a subsequence, we may assume thatW(xa;ya) is convergent inb(XX). LetW(xa;ya)!u.

We have:

c(u)ˆlim

a cW(x_a;y_a)ˆlim

a i_bi_b(x_a;y_a)ˆ(g₁;g₂):

p REMARK 4.4. Let R be an equivalence relation on a set X and let bd:X!bRbe a map defined bybd(x)ˆW(x;x). Suppose thatbX is the closure of the imageXunderbd. Then (bX;bd) is a compactification forX which is calledR-almost periodic compactification.

Acknowledgments. The authors are grateful to the referee for careful reading of the paper, and for providing several helpful suggestions for improvement in the article.

REFERENCES

[1] J. F. BERGLUND- H.D. JUNGHENN - P. MILNES: Analysis on Semigroups:

Function Spaces, Compactifications, Representations. Wiley, New York, 1989.

[2] BOCHNER - J. VON NEUMANN, Almost periodic functions in groups, Trans.

Amer. Math. Soc.37(1935), 21-50.

[3] H. BOHR,Zur Theorie der fastperiodischen Funcktionen, I, II, III, Acta Math.

45(1924), 29-127,46(1925), 101-214,47(1926), 237-281,

[4] A. BROWDER,Introduction to function algebras, W. A. Benjamin, New York, 1969.

[5] J. DIXMIER,C*-Algebras, North-Holland, Amsterdam, 1977.

[6] L. GILLMAN - M. JERISON, Rings of continuous functions, van Nostrand, Princeton, 1960.

[7] N. HINDMAN - D. STRAUSS, Algebra in the Stone-CÏech Compactification:

Theory and Applications, Walter de Gruyter, Berlin, 1998.

[8] P. MUHLY,Coordinates in operator algebra, (Book in preparation).

[9] A. L. T. PATERSON, Groupoids, inverse semigroups, and their operator algebras, Progress in Mathematics, 170. BirkhaÈuser, Boston, 1999.

[10] J. RENAULT,A groupoid approach to C-algebras, Lecture Notes in Math. 793, Springer, Berlin, 1980.

[11] S. ZAIDMAN,Almost-periodic functions in abstract spaces, Research Notes in Math. 126. Pitman, Boston, 1985.

Manoscritto pervenuto in redazione il 20 novembre 2012.