*-algebra, Morita equivalence">

Surveys in Mathematics and its Applications

ISSN 1842-6298
 Volume 1 (2006), 71 - 98

GROUPOID C*-ALGEBRAS

Mădălina Roxana Buneci

Abstract. The purpose of this paper is to recall the main ingredients of the construction of the C*-algebra of a groupoid (introduced by Renault in [19] and to collect some results on the independence of the C*-algebra on the choice of Haar system.

2000 Mathematics Subject Classification: 22A22, 43A05, 46L05, 46L52.
Keywords: locally compact Hausdorff groupoid, Haar system, C*-algebra, Morita equivalence.

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References

  1. C. Anantharaman-Delaroche, J. Renault, Amenable groupoids, Monographie de L'Enseignement Mathematique No 36, Geneve, 2000. MR1799683(2001m:22005). Zbl 0960.43003.

  2. W. Arveson, An invitation to C*-algebras, Springer-Verlag, New York, 1976. MR0512360(58 #23621). Zbl 0344.46123.

  3. L.G. Brown, P. Green, M. Rieffel, Stable isomorphism and strong Morita equivalence of C*-algebras, Pacific J. Math. 71 (1977), 349-363. MR0463928(57 #3866). Zbl 0362.46043.

  4. M. Buneci, C*-algebras associated to the transitive groupoids, An. Univ. Craiova Ser. Mat. Inform. 28 (2001), 79--92. MR1902038(2003b:46085). Zbl 1069.46510.

  5. M. Buneci, Haar systems and homomorphism on groupoids, Operator algebras and mathematical physics (Constanta, 2001), 35--49, Theta, Bucharest, 2003. MR2018222(2004j:22006).

  6. M. Buneci, The structure of the Haar systems on locally compact groupoids, Math. Phys. Electron. J. 8 (2002) Paper 4, 11 pp. (electronic). MR1978500(2004c:46102). Zbl 1023.46057.

  7. M. Buneci,Isomorphic groupoid C*-algebras associated with different Haar systems, New York J. Math. 11 (2005), 225-245. MR2154355. Zbl 1091.43001.

  8. M. Buneci , C*-algebras associated to groupoids with proper orbit space, Operator Theory 20 (Timisoara, 2004), 45-53.Theta Ser. Adv. Math., 6, Theta, Bucharest, 2006.

  9. A. Connes, Sur la theorie noncommutative de l'integration, Lecture Notes in Math. Springer-Verlag, Berlin 725 (1979) 19-143. MR0548112(81g:46090). Zbl 0412.46053.

  10. P. Hahn, Haar measure for measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 1-33. MR0496796(82a:28012). Zbl 0343.43003.

  11. P. Hahn, The regular representations of measure groupoids, Trans. Amer. Math. Soc. 242 (1978), 34-72. MR0496797(81f:46075). Zbl 0356.46055.

  12. G. Mackey, Induced representations of locally compact groups,Ann. of Math., 55 (1952), 101-139. MR0044536(13,434a). Zbl 0046.11601.

  13. P. Muhly, J. Renault and D. Williams, Equivalence and isomorphism for groupoid C*-algebras, J. Operator Theory 17 (1987), 3-22. MR0873460(88h:46123). Zbl 0645.46040.

  14. P. Muhly, Coordinates in operator algebra, (Book in preparation).

  15. W. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468. MR0355613(50 #8087). Zbl 0239.46062.

  16. A. Ramsay, Virtual groups and groups actions, Adv. in Math. 6 (1971), 253-322. MR0281876(43 #7590). Zbl 0216.14902.

  17. A. Ramsay, Topologies on measured groupoids, J. Funct. Anal. 47 (1982), 314-343. MR0665021(83k:22014). Zbl 0519.22003.

  18. A. Ramsay and M. E. Walter, Fourier-Stieltjes Algebras of locally compact groupoids, J. Funct. Anal. 148 (1997), 314-367. MR1469344(99a:22010). Zbl 0901.43002.

  19. J. Renault, A groupoid approach to C*-algebras, Lecture Notes in Math., Springer-Verlag, 793, 1980. MR0584266(82h:46075). Zbl 0433.46049.

  20. J. Renault,The ideal structure of groupoid crossed product algebras, J. Operator Theory, 25 (1991), 3-36. MR1191252(94g:46074). Zbl 0786.46050.

  21. M. Rieffel, Induced representations of C*-algebras, Adv. in Math. 13 (1974), 176-254. MR0353003(50 #5489). Zbl 0284.46040.

  22. M. Rieffel, Strong Morita equivalence of certain transformation group C*-algebras, Math. Ann. 222 (1976), 7-22. MR0419677(54 #7695). Zbl 0328.22013.

  23. A. K. Seda, A continuity property of Haar systems of measures, Ann. Soc. Sci. Bruxelles 89 IV (1975), 429-433. MR0402741(53 #6555). Zbl 0313.28018.

  24. A.K. Seda, Haar measures for groupoids, Proc. Royal Irish Acad. 76, Sec. A, no. 5, (1976), 25-36. MR0427598(55 #629). Zbl 0329.43002.

  25. A.K. Seda, On the continuity of Haar measure on topological groupoids, Proc. Amer. Math. Soc., 96 (1986), 115-120. MR0813822(87m:46146). Zbl 0593.28015.

  26. J. Westman, Nontransitive groupoid algebras, Univ. of California at Irvine, 1967.

  27. J. Westman, Harmonic analysis on groupoids, Pacific J. Math. 27 (1968), 621-632. MR0244443(39 #5758). Zbl 0167.44003.

Acknowledgement. This work was partially supported by the CEEX grant ET65/2005, contract no 2987/11.10.2005, and partially by CNCSIS grant A1065/2006 from the Romanian Ministry of Education and Research.

Mădălina Roxana Buneci
University Constantin Brâncuşi of Târgu-Jiu,
Bld. Republicii 1, 210152, Târgu-Jiu,
Romania.
e-mail: ada@utgjiu.ro
http://www.utgjiu.ro/math/mbuneci/

http://www.utgjiu.ro/math/sma