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Homogeneous C*-algebras whose spectra are tori

Part of: Selfadjoint operator algebras Fiber spaces and bundles

Published online by Cambridge University Press:  09 April 2009

Shaun Disney  and
Iain Raeburn
Shaun Disney
Affiliation:
School of Mathematics University of New South WalesP. O. Box 1 Kensington, N.S.W. 2033, Australia
Iain Raeburn
Affiliation:
School of Mathematics University of New South WalesP. O. Box 1 Kensington, N.S.W. 2033, Australia
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Abstract

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By a theorem of Fell and Tomiyama-Takesaki, an N-homogeneous C*-algebra with spectrum X has the form Γ(E) for some bundle E over X with fibre MN(C), and its isomorphism class is determined by that of E and its pull-backs f*E along homeomorphisms f of X. We describe the homogeneous C*-algebras with spectrum T2 or T3 by classifying the MN-bundles over Tk using elementary homotopy theory. We then use our results to determine the isomorphism classes of a variety of transformation group C*-algebras, twisted group C*-algebras and more general crossed products.

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 46L40: Automorphisms 55R15: Classification
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 38 , Issue 1 , February 1985 , pp. 9 - 39
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Backhouse, N. B., ‘Projective representations of space groups, II: factor systems’, Quart. J. Math 21 (1970), 277–295.CrossRefGoogle Scholar
[2]Backhouse, N. B. and Bradley, C. J., ‘Projective representations of space groups, I: translation groups’, Quart. J. Math. 21 (1970), 203–222.CrossRefGoogle Scholar
[3]Baggett, L. and Kleppner, A., ‘Multiplier representations of abelian groups’, J. Functional Analysis 14 (1973), 299–324.CrossRefGoogle Scholar
[4]Dixmier, J., C*-algebras (North-Holland, Amsterdam, 1977).Google Scholar
[5]Effros, E. G. and Hahn, F., ‘Locally compact transformation groups and C*-algebras’, Mem. Amer. Math. Soc 75 (1967).Google Scholar
[6]Fell, J. M. G., ‘The structure of algebras of operator fields’, Acta Math. 106 (1961), 233–280.CrossRefGoogle Scholar
[7]Høegh-Krohn, R. and Skjelbred, T., ‘Classification of C*-algebras admitting ergodic actions of the two-dimensional torus’, J. Reine Angew. Math. 328 (1981), 1–8.Google Scholar
[8]Husemoller, D., Fibre bundles, 2nd edition, (Springer-Verlag, Berlin and New York, 1975).Google Scholar
[9]Krauss, F. and Lawson, T. C., ‘Examples of homogeneous C*-algebras’, Mem. Amer. Math. Soc. 148 (1974), 153–164.Google Scholar
[10]Mackey, G. W., ‘Unitary representations of group extensions, I’, Acta Math. 99 (1958), 265–311.CrossRefGoogle Scholar
[11]Paulsen, V., ‘Weak compalence invariants for essentially n-normal operators’, Amer. J. Math. 101 (1979), 979–1006.CrossRefGoogle Scholar
[12]Pedersen, G. K. and Petersen, N. H., ‘Ideals in a C*-algebra’, Math. Scand. 27 (1970), 193–204.CrossRefGoogle Scholar
[13]Raeburn, I., ‘On group C*-algebras of bounded representation dimension’, Trans. Amer. Math. Soc. 272 (1982), 629–644.Google Scholar
[14]Raeburn, I. and Williams, D. P., ‘Pull-backs of C*-algebras and crossed products by certain diagonal actions’, Trans. Amer. Math. Soc., to appear.Google Scholar
[15]Rieffel, M. A., ‘Applications of strong Morita equivalence to transformation group C*-algebras’, Proc. Sympos. Pure Math. 38, (Operator algebras and applications) (1982), Part I, 299–310.CrossRefGoogle Scholar
[16]Rieffel, M. A., ‘The cancellation theorem for projective modules over irrational rotation algebras’, Proc. London Math. Soc. 47 (1983), 285–302.CrossRefGoogle Scholar
[17]Takesaki, M., ‘Covariant representations of C*-algebras and their locally compact automorphism groups’, Acta Math. 119 (1967), 273–303.CrossRefGoogle Scholar
[18]Tomiyama, J. and Takesaki, M., ‘Applications of fibre bundles to the certain class of C*-algebras’, Tohoku Math. J. 13 (1961), 498–522.CrossRefGoogle Scholar
[19]Woodward, L. M., ‘The classification of principal PUN-bundles over a 4-complex’, J. London Math. Soc. 25 (1982), 513–524.CrossRefGoogle Scholar
[20]Zeller-Meier, G., ‘Produits croisés d'une C*-algèbre par un groupe d'automorphismes’, J. Math. Pures Appl. 47 (1968), 101–239.Google Scholar