Correspondences and groupoids

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1 Proceedngs of the IX Fall Workshop on Geometry and Physcs, Vlanova la Geltrú, 2000 Publcacones de la RSME, vol. X, pp Correspondences and groupods 1 Marta Macho-Stadler and 2 Moto O uch 1 Departamento de Matemátcas, Unversdad del País Vasco-Euskal Herrko Unbertstatea 2 Department of Appled Mathematcs, Osaka Women s Unversty emals: mtpmastm@lg.ehu.es, ouch@appmath.osaka-wu.ac.jp Abstract Our defnton of correspondence between groupods (whch generalzes the noton of homomorphsm) s obtaned by weakenng the condtons n the defnton of equvalence of groupods n [5]. We prove that such a correspondence nduces another one between the assocated C*-algebras, and n some cases besdes a Kasparov element. We wsh to apply the results obtaned n the partcular case of K-orented maps of leaf spaces n the sense of [3]. Key words: Groupod, C*-algebra, correspondence, KK-group. MSC 2000: 46L80, 46L89 1 Introducton A. Connes ntroduced n [1] the noton of correspondence n the theory of Von Neumann algebras. It s a concept of morphsm, whch gves the well known noton of correspondence of C*-algebras. In [5], P.S. Mulhy, J.N. Renault and D. Wllams defned the noton of equvalence of groupods and showed that f two groupods are equvalent, then the assocated C*-algebras are Morta-equvalent. But, ths noton s too strong: here we gve a defnton of correspondence of groupods, byweakenng the condtons of equvalence n [5]. We prove that such a correspondence nduces another one (n the sense of A. Connes) between the assocated reduced C*-algebras. We show that a groupod homomorphsm satsfyng some addtonal condtons can be thought as a correspondence between them, and nduces a Kasparov element between the assocated C*-algebras. Our fundamental nterest s to apply these results to the study of folated spaces where, n many cases, homomorphsms between holonomy groupods appear (for example, n [2]).

2 2 Correspondences and groupods 2 Correspondences and groupods Let G ( =1, 2) be a second countable locally compact Hausdorff groupod. Let G (0) be the unt space and s,r : G G (0) the source and range map, respectvely (we do not assume that r, s are open). Let G,x = s 1 (x), for x G (0). Let Z be asecond countable locally compact Hausdorff space, ρ : Z G (0) 1 a contnuous and surjectve map, and the space G 1 Z = {(γ 1,z) G 1 Z : s 1 (γ 1 )=ρ(z)}. Defnton 1 A left acton of G 1 on Z s a contnuous and surjectve map Φ:G 1 Z Z, noted Φ(γ 1,z)=γ 1.z, wth the followng propertes: 1) ρ(γ 1.z) =r 1 (γ 1 ), for (γ 1,z) G 1 Z, 2) γ 1.(γ 1.z) =(γ 1 γ 1).z, when both sdes of the equalty are defned, 3) ρ(z).z = z, for each z Z, and we say thus that Z s a left G 1 -space. Z s called proper, when the map Φ 1 : G 1 Z Z Z defned by Φ 1 (γ 1,z)=(γ 1.z, z), s proper. In the same manner, we defne a rght G 2 -acton on Z, usng a contnuous and surjectve map σ : Z G (0) 2 and Z G 2 = {(z,γ 2 ) Z G 2 : r 2 (γ 2 )=σ(z)}. Defnton 2 Let G 1 and G 2 be second countable locally compact Hausdorff groupods and Z a second countable locally compact Hausdorff space. The space Z s a correspondence from G 1 to G 2, when: 1) there s a left proper G 1 -acton on Z and a rght proper G 2 -acton on Z, and they commute, 2) ρ : Z G (0) 1 s open and nduces a bjecton of Z/G 2 onto G (0) 1. The dfference wth the defnton of equvalence n [5], s that here we do not suppose that the actons are free, we do not assume that σ s open and above all, we do not suppose that σ nduces a bjecton of G 1 \Z onto G (0) 2. If V Z, Sat 2 (V )={z.γ 2 Z : z V and (z,γ 2 ) Z G 2 } s ts saturaton wth respect to the G 2 -acton and (2) of defnton 2 s true, then Sat 2 (V )=ρ 1 ρ(v ) and the quotent map Z Z/G 2 s open. Moreover, f the G 2 -acton s proper, then Z/G 2 s a locally compact Hausdorff space. Defnton 3 Let A and B be C*-algebras. The couple (E,φ) s a correspondence from A to B, f t satsfes the followng propertes:

3 Marta Macho-Stadler and Moto O uch 3 1) E s a rght Hlbert B-module, 2) φ s a -homomorphsm from A nto L B (E), the set of bounded adjontable operators on E. If φ(a) K B (E) (the closure of the lnear span of {θ ξ,η } ξ,η E, where θ ξ,η L B (E) sdefned by θ ξ,η (ζ) =ξ η, ζ, for ζ E), then (E,φ,0) s a Kasparov module for trvally graded C*-algebras (A, B) and gves an element [E] of KK(A, B). Note that each -homomorphsm between C*-algebras nduces a correspondence between them. For {1, 2}, let λ be a rght Haar system of G (ths condton mples that r and s are open). Let C c (G )bethe -algebra of compactly supported contnuous functons, where the product and the nvoluton are defned by: 1 (ab)(γ )= a(γ γ )b(γ )dλ s (γ ) (γ ) and a (γ )=a(γ 1 ), G for a, b C c (G ) and γ G.Forx G (0),wedefne a representaton π,x of C c (G )onl 2 (G,x,λ x)by: (π,x (a)ζ)(γ )= a(γ γ 1 )ζ(γ )dλ x(γ ) G for a C c (G ), ζ L 2 (G,x,λ x) and γ G,x. We defne the reduced norm by a = sup x G (0) π,x (a), and the reduced groupod C*-algebra Cr (G )sthe completon of C c (G )by the reduced norm. Theorem 4 (see [4]) Let (G,λ ) ( {1, 2}) beasecond countable locally compact Hausdorff groupod wth a rght Haar system λ and Z acorrespondence from G 1 to G 2. There exsts a correspondence from C r (G 2 ) to C r (G 1 ). 3 Homomorphsms of groupods Let G 1 and G 2 be as n the prevous secton and let f be a contnuous homomorphsm of G 1 onto G 2. We denote by f (0) the restrcton of f to G (0) 1, whch s a map onto G (0) 2. The kernel of f, H = {γ 1 G 1 : f(γ 1 ) G (0) 2 }, s a closed subgroupod of G 1 and we have H (0) = G (0) 1. There s a natural

4 4 Correspondences and groupods rght acton of H on G 1, whch s proper snce H s closed. We defne the map (r, s) H : H H (0) H (0) by (r, s) H (γ) =(r H (γ),s H (γ)), for γ H, where r H and s H are the range and source map of H, respectvely. Then: Theorem 5 (see [4]) Let G 1 and G 2 be second countable locally compact Hausdorff groupods, let f be acontnuous homomorfsm of G 1 onto G 2 and let H be the kernel of f. Suppose that the followng propertes are satsfed: (C1) the quotent map q H : G 1 G 1 /H s open, (C2) r 1 : G 1 G (0) 1 s open, (C3) (r, s) H : H H (0) H (0) s proper, (C4) for each x G (0) 1, f(g 1,x) =G 2,f(x), (C5) f : G 1 G 2 s open, and (C6) f 0 : G (0) 1 G (0) 2 s locally one-to-one. Then, G 1 /H s a correspondence from G 1 to G 2. An homomorphsm of groupods does not nduce, n general, an homomorphsm between the assocated C*-algebras, but the followng result holds: Theorem 6 (see [4]) Let (G,λ ) be asecond countable locally compact Hausdorff groupod wth a rght Haar system λ for =1, 2, and let f be acontnuous homomorphsm of G 1 onto G 2. Suppose that the condtons (C1) to (C6), and the followng condton are satsfed: (C7) f 0 : G (0) 1 G (0) 2 s proper. Then, there s a correspondence (E,φ) from C r (G 2 ) to C r (G 1 ), such that φ(c r (G 2 )) K C r (G 1 )(E). Thus, (E,φ,0) s a Kasparov module for the couple (C r (G 2 ),C r (G 1 )), and we obtan an element of KK(C r (G 2 ),C r (G 1 )). 4 Some examples Let G ( =1, 2), f and H be as n Theorem 5. Suppose that they satsfy the condtons (C1) to (C6). Set Z = G 1 /H. Denote by λ a rght Haar system of G.Itfollows from Theorems 4 and 5 that we have a correspondence from Cr (G 2 )tocr (G 1 ). Denote by (E,φ) the correspondence constructed n the proof of Theorem 4. If the condton (C7) s satsfed, then (E,φ,0) s a

5 Marta Macho-Stadler and Moto O uch 5 Kasparov module and gves an element of KK(C r (G 2 ),C r (G 1 )) by Theorem 6. In ths secton, we study some examples where groupods are topologcal spaces, topologcal groups and transformaton groups, respectvely. 4.1 Topologcal Spaces Let X beatopologcal space and suppose that G s the trval groupod X, {1, 2}. Then, f : X 1 X 2 s contnuous and surjectve and C r (G )sthe commutatve C*-algebra C 0 (X )ofcontnuous functons vanshng at nfnty. Remark that f (0) = f, H = X 1 and X 1 /H = X 1. We have E = C 0 (X 1 ) and thus, φ s the -homomorphsm φ : C 0 (X 2 ) M(C 0 (X 1 )) (M(C 0 (X 1 )) s the multpler algebra of C 0 (X 1 )), defned by φ(b) =b(f(x 1 )), for b C 0 (X 2 ) and x 1 X 1.If(C7) s satsfed, then f s proper and φ(c 0 (X 2 )) C 0 (X 1 ). 4.2 Topologcal Groups Let Γ beatopologcal group and suppose that G =Γ. Then, f :Γ 1 Γ 2 s an epmorphsm and H = Ker(f). By (C5), f s open. Therefore, Γ 1 /H and Γ 2 are somorphc topologcal groups, and thus f can be thought as the quotent map f :Γ 1 Γ 1 /H. Snce G (0) = {e } (e s the unt of Γ ), f (0) s trval and (C7) s always satsfed. Moreover, H s a compact group by (C3). We defne λ as a rght Haar measure on Γ. 4.3 Transformaton groups Let Γ be a topologcal group, X a rght Γ -space and G = X Γ. The groupod structure of G s defned by r (x,g )=x, s (x,g )=x g and (x,g )(x g,g )=(x,g g ), where we dentfy G(0) wth X. Moreover, we suppose that there s a surjectve map f (0) : X 1 X 2 and an epmorphsm ϕ :Γ 1 Γ 2, such that f(x, g) =(f (0) (x),ϕ(g)) and f (0) (xg) =f (0) (x)ϕ(g). By (C5), f (0) and ϕ are open maps. If Ξ = Ker(ϕ), we dentfy Γ 1 /Ξ wth Γ 2. Then, ϕ s the quotent map. We have H = X 1 Ξ and Z = X 1 Γ 2. The condton (C3) s satsfed f and only f the Ξ-acton s proper. We defne ρ : Z X 1 and σ : Z X 2 by ρ(x 1,g 2 )=x 1 and σ(x 1,g 2 )= f (0) (x 1 )g 2. The G 1 -acton and the G 2 -acton on Z are defned respectvely by (x 1 g1 1,g 1).(x 1,g 2 )=(x 1 g1 1,g 1.g 2 ) and (x 1,g 2 ).(f (0) (x 1 )g 2,g 3 )=(x 1,g 2 g 3 ), for (x 1,g 2 ) Z, (x 1 g1 1,g 1) G 1 and (f (0) (x 1 )g 2,g 3 ) G 2.

6 6 Correspondences and groupods 5 Further research If (M, F ), {1, 2}, are folated manfolds and f : M 1 /F 1 M 2 /F 2 sakorented morphsm of leaf spaces (.e., a correspondence between ts holonomy groupods), we look for an element of the Kasparov group KK(C r (G 1 ),C r (G 2 )), where G s the holonomy groupod of (M, F ). At present, we study the case of transversely affne folatons, snce ther holonomy groupods are Hausdorff. But holonomy groupods are non Hausdorff n many cases of nterestng folated spaces. Thus, we ntend to extend our results to non Hausdorff groupods. Acknowledgments Ths work has been partally supported by UPV EA 005/99 References [1] A. Connes, Non commutatve Geometry, Academc Press, [2] G. Hector and M. Macho Stadler, Isomorphsme de Thom pour les feulletages presque sans holonome, Comptes Rend. Acad. Sc. 325 (1997) [3] M. Hlsum and G. Skandals, Morphsmes K-orentés d espaces de feulles et fonctoralté en théore de Kasparov (d après une conjecture d A. Connes), Ann. Sc. Ec. Norm. Sup. 20 (1987) [4] M. Macho Stadler and M. O uch, Correspondence of groupod C*-algebras, Journal of Operator Theory 42 (1999) [5] P.S. Mulhy, J. Renault and D.P. Wllams, Equvalence and somorphsm for groupod C*-algebras, J. Operator Theory 17 (1987) 3 22.