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Representations of triangular subalgebras of groupoid C*-algebras

Published online by Cambridge University Press:  09 April 2009

Paul S. Muhly
Affiliation:
Department of Mathematics University of IowaIowa City, IA 52242, USA
Baruch Solel
Affiliation:
Department of Mathematics Technion-Israel Institute of Technology Haifa 32000, Israel
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Abstract

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We investigate the invariant subspace structure of subalgebras of groupoid C*-algebras that are determined by automorphism groups implemented by cocycles on the groupoids. The invariant subspace structure is intimately tied to the asymptotic behavior of the cocycle.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1996

References

[1]Arveson, W. B., ‘On groups of automorphisms of operator algebras’, J. Funct. Anal. 15 (1974), 217243.CrossRefGoogle Scholar
[2]Arveson, W. B., ‘Subalgebras of C*-algebras I’, Acta Math. 123 (1969), 141224.CrossRefGoogle Scholar
[3]Arveson, W. B., ‘Subalgebras of C*-algebras II’, Acta Math. 128 (1972), 271308.CrossRefGoogle Scholar
[4]Arveson, W. B. and Josephson, K., ‘Operator algebras and measure preserving automorphisms II’, J. Funct. Anal. 4 (1969), 100134.CrossRefGoogle Scholar
[5]Doubilet, P., Rota, G. -C., and Stanley, R., ‘On the foundations of combinatorial theory (VI): The idea of a generating function’, in: Proc. Sixth Berkeley Symposium in Math., Stat. and Prob., Vol. II (1971), pp. 267318.Google Scholar
[6]Feldman, J. and Moore, C., ‘Ergodic equivalence relations, cohomology and von Neumann algebras I’, Trans. Amer. Math. Soc. 234 (1977), 289324.CrossRefGoogle Scholar
[7]Feldman, J. and Moore, C., ‘Ergodic equivalence relations, cohomology and von Neumann algebras II’, Trans. Amer. Math. Soc. 234 (1977), 325359.CrossRefGoogle Scholar
[8]Hopenwasser, A. and Power, S. C., ‘Classification of limits of triangular matrix algebras’, Proc. Edingburgh Math. Soc. 36 (1992), 107121.CrossRefGoogle Scholar
[9]Kawamura, S. and Tomiyama, J., ‘On subdiagonal algebras associated with flows in operator algebras’, J. Math. Soc. Japan 29 (1977), 7390.CrossRefGoogle Scholar
[10]Kumjian, A., ‘On C*-diagonals’, Canad. J. Math. 38 (1986), 9691008.CrossRefGoogle Scholar
[11]Loebl, R. I. and Muhly, P. S., ‘Analyticity and flows in von Neumann algebras’, J. Funct. Anal. 29 (1978), 214252.CrossRefGoogle Scholar
[12]Muhly, P. S., Qiu, C. and Solel, B., ‘Coordinates, nuclearity and spectral subspace in operator algebra’, J. Operator Theory 26 (1991), 313332.Google Scholar
[13]Muhly, P. S., Saito, K.-S. and Solel, B., ‘Coordinates for triangular operator algebras’, Ann. of Math. 127 (1988), 245278.CrossRefGoogle Scholar
[14]Muhly, P. S., Saito, K.-S. and Solel, B., ‘Coordinates for triangular operator algebras II’, Pacific J. Math. 137 (1989), 335369.CrossRefGoogle Scholar
[15]Muhly, P. S. and Solel, B., ‘On triangular subalgebras of groupoid C*-algebras’, Israel. J. Math. 71 (1990), 257273.CrossRefGoogle Scholar
[16]Muhly, P. S., and Solel, B., ‘Dilations and commutant lifting for subalgebras of groupoid C*-algebras’, Internat. J. Math. 5 (1994), 87123.CrossRefGoogle Scholar
[17]Muhly, P. S., and Solel, B., ‘Hilbert modules over operator algebras’, Mem. Amer. Math. Soc. vol. 117, #559 (1995).Google Scholar
[18]Olesen, D., ‘On spectral subspaces and their applications to automorphism groups’, Sympos. Math. 20 (1976), 253296.Google Scholar
[19]Orr, J. L. and Peters, J. R., ‘Some representations of TAF algebras’, Pacific J. Math. 167 (1995), 129161.CrossRefGoogle Scholar
[20]Pedersen, G. K., C*-algebras and their automorphism groups, London Math. Soc. Monograph 14, (Academic Press, New York, 1979).Google Scholar
[21]Peters, J. R., Poon, Y. T. and Wagner, B. H., ‘Triangular AF algebras’, J. Operator Theory 23 (1990), 81114.Google Scholar
[22]Peters, J. R., Poon, Y. T. and Wagner, B. H., ‘Analytic TAF algebras’, Canad. J. Math. 45 (1993), 10091031.CrossRefGoogle Scholar
[23]Peters, J. R. and Wagner, B. H., ‘Triangular AF algebras and nest subalgebras of UHF algebras’, J. Operator Theory 25 (1991), 79124.Google Scholar
[24]Power, S. C., Limit algebras: an introduction to subalgebras of C*-algebras, Pitman Res. Notes Math. Ser. 278 (Longman Sci. and Tech., 1992).Google Scholar
[25]Ramsay, A., ‘Nontransitive quasi-orbits in Mackey's analysis of group extensions’, Acta Math. 137 (1976), 1748.CrossRefGoogle Scholar
[26]Renault, J., A groupoid approach to C*-algebras, Lecture Notes in Math. 793 (Springer, Berlin, 1980).CrossRefGoogle Scholar
[27]Renault, J., Représentation des produits croisés d'algèbres de groupoîdes, J. Operator Theory 18 (1987), 6797.Google Scholar
[28]Schmidt, K., Cocycles on ergodic transformation groups, McMillan Lect. in Math. 1 (The Macmillan Company of India, 1977).Google Scholar
[29]Solel, B., Applications of the asymptotic range to analytic subalgebras of groupoid C*-algebras, Ergodic Theory Dynamical Systems 12 (1992), 341358.CrossRefGoogle Scholar