2 MADALINA ROXANA BUNECI subset G (2) G G (called the set of composable pars), and two maps: h (x y)! xy : G (2)! G (product map) x! x ;1 [: G! G] (nv

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1 An applcaton of Mackey's selecton lemma Madalna Roxana Bunec Abstract. Let G be a locally compact second countable groupod. Let F be a subset of G (0) meetng each orbt exactly once. Let us denote by df the restrcton of the doman map to G F and by r 0 the restrcton of the range map to the sotropy group bundle of G. We shall prove that f df s open, then r 0 s open and df has a regular Borel cross secton. Conversely, we shall prove thatfr 0 s open and df admts a regular cross secton (a rght nverse whch carres each compact subset of G (0) nto a relatvely compact subset of G F ), then df s open. We shall also prove that, f df s open, then F s a closed subset of G (0), and the orbt space G (0) =G s a proper space. If F s closed and regular (the ntersecton of F wth the saturated of any compact subset of G (0) s relatvely compact) and G (0) =G s proper, then df s open. AMS 2000 Subject Classcaton: 22A22, 43A05, 54D45, 37A55 Key words: locally compact groupod, regular cross secton, orbt space to appear n Studa Unverstats Babes-Bolya, Matematca vol. L(4) (2005), Introducton We shall consder a locally compact groupod G and a set F contanng exactly one element from each orbt of G. We shall study the connecton between the openness of d F, the restrcton of the doman map to G F, and the exstence of a regular cross secton of d F (a rghtnverse whch carres each compact subset of G (0) nto a relatvely compact subset of G F ). The motvaton for studyng the map d F comes from the fact that f F s closed and d F s open, then G and G F F are (Morta) equvalent locally compact groupods (n the sense of Denton 2.1/p. 6[4]). A result of Paul Muhly, Jean Renault and Dana Wllams states that the C -algebras assocated to (Morta) equvalent locally compact second countable groupods are strongly Morta equvalent (Theorem 2.8/p. 10 [4]). Also the noton of topologcal amenablty snvarant under the equvalence of groupods (Theorem 2.2.7/p. 50 [1]). Consequently, ff s closed and d F s open, then G and the bundle group G F F have strongly Morta equvalent C -algebras. Also, the equvalence of G and G F F mples that G s amenable f and only f each sotropy group G u u s amenable. For establshng notaton, we nclude some dentons that can be found n several places (e.g. [5]). A groupod s a set G, together wth a dstngushed Ths work s supported by the MEC-CNCSIS grant At 127/2004 (33346/ ). 1

2 2 MADALINA ROXANA BUNECI subset G (2) G G (called the set of composable pars), and two maps: h (x y)! xy : G (2)! G (product map) x! x ;1 [: G! G] (nverse map) such that the followng relatons are satsed: (1) If (x y) 2 G (2) and (y z) 2 G (2), then (xy z) 2 G (2), (x yz) 2 G (2) and (xy) z = ; x (yz). (2) x ;1 ;1 = x for all x 2 G. (3) For all x 2 G, ; x x ;1 2 G (2), and f (z x) 2 G (2),then(zx) x ;1 = z. (4) For all x 2 G, x ;1 x 2 G (2), and f (x y) 2 G (2),thenx ;1 (xy) =y. The maps r and d on G, dened by the formulae r (x) =xx ;1 and d (x) =x ;1 x, are called the range and the source maps. It follows easly from the denton that they have a common mage called the unt space of G, whch s denoted G (0). Its elements are unts n the sense that xd (x) =r (x) x = x. It s useful to note that a par (x y) les n G (2) precsely when d (x) = r (y), and that the cancellaton laws hold (e.g. xy = xz y = z). The bers of the range and the source maps are denoted G u = r ;1 (fug) and G v = d ;1 (fvg), respectvely. More generally, gven the subsets A, B G (0), we dene G A = r ;1 (A), G B = d ;1 (B) and G A = B r;1 (A) \ d ;1 (B). G A A becomes ; a groupod (called the reducton of G to A) wth the unt space A, fwe dene GA ; A (2) = G (2) \ G A A A GA. For each unt u, G u u s a group, called sotropy group at u. The group bundle fx 2 G : r (x) =d (x)g s denoted G 0, andscalledthesotropy group bundle of G. The relaton u~v G u v 6= s an equvalence relaton on G(0). Its equvalence classes are called orbts and the orbt of a unt u s denoted [u]. Let R =(r d)(g) =f(r (x) d(x)) x 2 Gg be the graph of the equvalence relaton nduced on G (0).The quotent space for ths equvalence relaton s called the orbt space of G and denoted G (0) =G. A topologcal groupod conssts of a groupod G and a topology compatble wth the groupod structure. We are exclusvely concerned wth topologcal groupods whch are locally compact Hausdor. The Borel sets of a topologcal space are taken to be the -algebra generated by the open sets. 2. Necessary and sucent condtons for the openness of d F. Defnton 1. Let X Y be two topologcal spaces. A cross secton of a map f : X! Y s a functon : Y! X such that f ( (y)) = y for all y 2 Y. We shall say that the cross secton s regular f (K) has compact closure n X for each compact set K n Y. We shall need the followng lemma proved by Mackey (Lemma 1.1/p. 102 [3]): Lemma 1. If X and Y are second countable, locally compact spaces, and f : X! Y s a contnuous open functon onto Y, then f has a Borel regular cross secton.

3 AN APPLICATION OF MACKEY'S SELECTION LEMMA 3 Proposton 1. Let G be alocally compact groupod. Let F be a subset of G (0) meetng each orbt exactly once. Let us dene the functon e : G (0)! G (0) by e (u) =F \ [u] u2 G (0) If the map d F : G F! G (0) d F (x) = d (x), s open, then the functon e s contnuous and F s a closed subset of G (0). Proof. Let (u ) be a net convergng to u n G (0). Let x 2 G be such that r (x) =e (u) andd (x) =u. Snce (u ) converges to d F (x) and d F s an open map, we may pass to a subnet and assume that there s a net (x ) convergng to x n G F such thatd F (x )=u. It s easy to see that r (x )=e(u )( r (x ) 2 F and r (x ) 2 [d (x )]=[u ]). Thus e (u )=r(x ) converges to r (x) =e (u). Snce G (0) s Hausdor, F s closed n G (0), beng the mage of the map e whose square s tself. Proposton 2. Let G be a locally compact groupod. Let F be a subset of G (0) meetng each orbt exactly once. If the map d F : G F! G (0) d F (x) =d (x), s open, then graph R = f(r (x) d(x)) x 2 Gg of the equvalence relaton nduced on G (0) s closed n G (0) G (0), and the map (r d) : G! R, (r d)(x) = (r (x) d(x)) s open, where R s endowed wth the product topology nduced from G (0) G (0). Proof. Let us dene the functon e : G (0)! G (0) by e (u) =F \ [u] u2 G (0). By Proposton 1, the functon e s contnuous. Let ((u v )) be a net n R whch converges to (u v) n G (0) G (0) (wth respect to wth the product topology ). Then (u ) converges to u, (v ) converges to v, andu ~ v for all. We have lm e (u ) = e (u) lm e (v ) = e (v) because e s contnuous. On the other hand, the fact that u ~ v for all mples that e (u )=e(v ) for all. Hence e (u) =e (v), or equvalently, u~v. Therefore (u v) 2 R. Let us prove that the map (r d) : G! R, (r d)(x) = (r (x) d(x)) s open, where R s endowed wth the product topology nduced from G (0) G (0). Let x 2 G, and let ((u v )) be a net n R convergng to (r d)(x). Then (u ) converges to r (x), (v ) converges to d (x), and u ~ v for all. Let s 2 G be such thatr (s) =e (r (x)) and d (s) =r (x) and let t = sx. Obvously, s t 2 G F and lm u = r (x) =d (s) lm v = d (x) =d (sx) =d (t). Snce d F s an open map, we may pass to subnets and assume that there s a net (s ) convergng to s n G F and there s a net (t ) convergng to t n G F such

4 4 MADALINA ROXANA BUNECI that d F (s )=u and d F (t )=v. The fact that e (u ) s the only element of F, whch s equvalent tou ~ v, mples that r (s )=e (u )=e (v )=r (t ). We have lm ; s ;1 t = s ;1 t = x r s ;1 Therefore the map (r d)sopen t = d (s )=u d ; s ;1 t = d (t )=v Corollary 1. Let G be a locally compact groupod havng open range map. Let F be a subset of G (0) meetng each orbt exactly once. If the map d F : G F! G (0) d F (x) =d (x), sopen, then the orbt space G (0) =G s proper. Proof. The fact that G (0) =G s a proper space means that G (0) =G s Hausdor andthemap(r d):g! R, (r d)(x) =(r (x) d(x)) s open, where R s endowed wth the product topology nduced from G (0) G (0). Let us note that the quotent map : G (0)! G (0) =G s open (because the range map of G s open). Snce the graph R of the equvalence relaton s closed n G (0) G (0), t follows that G (0) =G s Hausdor. Lemma 2. Let G be a locally compact groupod havng open range map. Let F be a subset of G (0) meetng each orbt exactly once. If the map d F : G F! G (0) d F (x) =d (x), sopen, then F and G (0) =G are homeomorphc spaces. Proof. Let : G (0)! G (0) =G be the quotent map. We prove that the map F : F! G (0) =G F (x) = (x) s a homeomorphsm. It suces to prove that F s an open map (because F s one-to-one from F onto G (0) =G). Let u 2 F and (_u ) be a net convergng to (u) n G (0) =G. Snce d F s open, we may pass to a subnet and assume that there s a net (x ) convergng to u n G F such that (d F (x )) = _u. Then (r (x )) s a net n F whch converges to u. Remark 1. Let G be a locally compact groupod. If the map (r d) : G! R s open (where R = f(r (x) d(x)) x2 Ggs endowed wth the product topology nduced from G (0) G (0) ), then the map r 0 : G 0! G (0) r 0 (x) = r (x), s open, where s the sotropy group bundle of G. G 0 = fx 2 G : r (x) =d (x)g, Proposton 3. Let G be a locally compact second countable groupod. Let F be a subset of G (0) meetng each orbt exactly once. If the map d F : G F! G (0) d F (x) =d (x), sopen, then d F has Borel regular cross secton. Proof. If d F s open, then accordng to Proposton 1, F s a closed subset of G (0). Therefore G F s a locally compact space, and we may apply Lemma 1. We shall need a system of measures f u v (u v) 2 (r d)(g)g satsfyng the followng condtons: 1. supp (v u )=G u v for all u ~ v. 2. sup R v u (K) < 1 for R all compact K G. u v 3. f (y) d r(x) v (y) = f (xy) d d(x) v (y) for all x 2 G and v ~ r (x).

5 AN APPLICATION OF MACKEY'S SELECTION LEMMA 5 In Secton 1 of [6] Jean Renault constructs a Borel Haar system for G 0. One way todo ths s to choose a functon F 0 contnuous wth condtonally support, whch s nonnegatve and equal to 1 at each u 2 G (0) : Then for each u 2 G (0) choose a left Haar measure u u on G u u so the ntegral of F R 0 wth respect to u u s 1. Renault denes v u = xv v f x 2 G u v (where xv v (f) = f (xy) dv v (y) as usual). If z s another element ng u v,thenx ;1 z 2 G v v, and snce v v s a left Haar measure on G v v, t follows that v u s ndependent ofthechoce of x. If K s a compact subset of G, then sup v u (K) < 1. We obtan another constructon of a system a measures wth u v the above propertes f n the proof of Theorem 8/p. 331[2] we replace the regular cross secton of G u! d G (0) (n the transtve case) wth a regular cross secton of G F! d G (0), where F s a subset of G (0) meetng each orbt exactly once. Lemma 3. Let G be a locally compact groupod. Let F be a subset of G (0) meetng each orbt exactly once and let us denote e (u) the unque element of F equvalent to u. If the map u 7! f (y) d e(u) u (y) h : G (0)! C s contnuous for any contnuous functon wth compact support, f : G! C, then the map s open. d F : G F! G (0), d F (x) =d (x). Proof. Let x 0 2 G F and let U be a nonempty compact neghborhood of x 0. Choose a nonnegatve contnuous functon, f on G,wthf(x 0 ) > 0andsupp (f) U. Let W be the set of unts u wth the property that e(u) u (f) > 0. Then W s an open neghborhood of u 0 = d(x 0 ) contaned n d F (U). Proposton 4. Let G be alocally compact second countable groupod. Let F be a subset of G (0) contanng exactly one element from each orbt of G, and let us denote e (u) the unque element of F equvalent to u. Let us assume that the map r 0 : G 0! G (0) r 0 (x) =r (x) s open, where G 0 s the sotropy group bundle of G. If the map d F : G F! G (0) d F (x) =d (x), hasaregular cross secton, then for each contnuous wth compact support functon f : G! C, the map s contnuous on G. u! f (y) d e(u) u (y) Proof. By Lemma 1.3/p. R 6[6], for each f : G! C contnuous wth compact support, the functon u! f (y) du u (y) : G (0)! C s contnuous. Let (u ) be a sequence n G (0) convergng to u. Let x = (u ) ;1 : Snce s regular, t follows that (x ) has a convergent subsequence n G F. Let x be the lmt of ths subsequence. Let f : G! C beacontnuous functon wth compact support and let g be acontnuous extenson on G of y! f (xy) : G d(x)! C. Let K be the compact set fx x =1 2 ::g ;1 supp (f) [ supp (g) \r ;1 (fd (x) d(x ) =1 2 :::g). We

6 6 MADALINA ROXANA BUNECI have = = f (y) d e(u) u (y) ; f (xy) d d(x) g (y) d d(x) d(x) (y) ; g (y) d d(x) d(x) (y) ; + g (y) d d(x) g (y) d u u (y) ; f (y) du e(u) (y) d(x) (y) ; f (x y) d d(x) d(x) (y) f (x y) d d(x) d(x) (y) d(x) (y) ; g (y) d d(x) d(x) (y) + f (x y) d d(x) d(x) (y) g (y) d u u (y) + + sup jg (y) ; f (x y)j y2g u u u (K) u A compactness argument shows that sup y2g u u jg (y) ; f (x y)j converges to 0. Also R R ;R g (y) d d(x) d(x) (y) f (x y) d d(x) d(x) (y) converges to 0, because the functon u! f (y) du u (y) scontnuous on G (0). Hence converges to 0: f (y) d e(u) u (y) ; f (y) d e(u) u (y) Corollary 2. Let G be a locally compact second countable groupod. Let F be a subset of G (0) meetng each orbt exactly once. If the restrcton r 0 of the range map to the sotropy group bundle G 0 of G s open, and f the map d F : G F! G (0) d F (x) =d (x), hasaregular cross secton, then d F s an open map. Theorem 1. Let G be alocally compact second countable groupod. Let F be a subset of G (0) meetng each orbt exactly once, and let d F : G F! G (0) be the map dened by d F (x) = d (x) for all x 2 G F. If d F s open then d F admts a Borel regular cross secton. If the restrcton r 0 of the range map to the sotropy group bundle G 0 of G s open and f d F admts a regular cross secton, then d F s an open map. Proof. If d F s an open map, then, accordng Proposton 3, d F has a regular cross secton. Conversely, fd F admts a regular cross secton, then applyng Proposton 4 and Lemma 3, t follows that d F s open. Remark 2. Let us assume that G (0) =G s proper. There s a regular Borel cross secton 0 of the quotent map : G (0)! G (0) =G. Let us assume that F = 0 ; G (0) =G s closed n G (0). Then the functon e : G (0)! G (0) dened by e (u) =F \ [u] s contnuous. If 1 : R! G s regular Borel cross secton of (r d), then : G (0)! G F, (u) = 1 (e (u) u) s a Borel regular cross secton of d F. Therefore s that case d F s open.

7 AN APPLICATION OF MACKEY'S SELECTION LEMMA 7 References [1] C. Anantharaman-Delaroche, J. Renault, Amenable groupods, Monographe de L'Ensegnement Mathematque No 36, Geneve, [2] M. Bunec, Consequences of Hahn structure theorem for the Haar measure, Math. Reports, 4(54) (2002), no. 4, [3] G. Mackey, Induced representatons of locally compact groups. I, Ann. of Math., 55(1952), [4] P. Muhly, J. Renault and D. Wllams, Equvalence and somorphsm for groupod C*-algebras, J. Operator Theory 17(1987), [5] J. Renault, A groupod approach to C - algebras, Lecture Notes n Math., Sprnger-Verlag, 793, [6] J. Renault,The deal structure of groupod crossed product algebras, J. Operator Theory, 25(1991),3-36. Unversty Constantn Br^ancus of T^argu-Ju, Bulevardul Republc, Nr. 1, T^argu-Ju, Romana. e-mal: ada@utgju.ro