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Naturality and induced representations

Published online by Cambridge University Press:  17 April 2009

Siegfried Echterhoff ,
S. Kaliszewski ,
John Quigg  and
Iain Raeburn
Siegfried Echterhoff
Affiliation:
Fachbereich Mathematik und Informatik, University of Münster, 48149 Münster, Germany, e-mail: echters@uni-muenster.de
S. Kaliszewski
Affiliation:
Department of Mathematics, Arizona State University, Tempe AZ 85287, United States of America, e-mail: quigg@math.la.asu.edu
John Quigg
Affiliation:
Department of Mathematics, Arizona State University, Tempe AZ 85287, United States of America, e-mail: kaz@math.la.asu.edu
Iain Raeburn
Affiliation:
Department of Mathematics, University of Newcastle, New South Wales 2308, Australia, e-mail: iain@maths.newcastle.edu.au
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Abstract

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We show that induction of covariant representations for C*-dynamical systems is natural in the sense that it gives a natural transformation between certain crossed-product functors. This involves setting up suitable categories of C*-algebras and dynamical systems, and extending the usual constructions of crossed products to define the appropriate functors. From this point of view, Green's Imprimitivity Theorem identifies the functors for which induction is a natural equivalence. Various special cases of these results have previously been obtained on an ad hoc basis.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 3 , June 2000 , pp. 415 - 438
Copyright
Copyright © Australian Mathematical Society 2000

References

REFERENCES

[1]Curto, R., Muhly, P. and Williams, D., ‘Cross products of strongly Morita equivalent C *-algebras, Proc. Amer. Math. Soc. 90 (1984), 528–530.Google Scholar
[2]Combes, F., ‘Crossed products and Morita equivalence, Proc. London Math. Soc. 49 (1984), 289–306.CrossRefGoogle Scholar
[3]Connes, A., Noncommutative geometry (Academic Press, San Diego, 1994).Google Scholar
[4]Echterhoff, S., ‘Morita equivalent twisted actions and a new version of the Packer-Raeburn stabilization trick, J. London. Math. Soc. 50 (1994), 170–186.CrossRefGoogle Scholar
[5]Echterhoff, S. and Raeburn, I., ‘Multipliers of imprimitivity bimodules and Morita equivalence of crossed products, Math. Scand. 76 (1995), 289–309.CrossRefGoogle Scholar
[6]Echterhoff, S. and Raeburn, I., ‘The stabilisation trick for coactions, J. reine angew. Math. 470 (1996), 181–215.Google Scholar
[7]Green, P., ‘The local structure of twisted covariance algebras, Acta Math. 140 (1978), 191–250.CrossRefGoogle Scholar
[8]Kasparov, G.G., ‘Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988), 147–201.CrossRefGoogle Scholar
[9]Kasparov, G.G., ‘K-theory, group C *-algebras, and higher signatures (conspectus)’, in Novikov conjectures, index theorems, and rigidity. Vol. 1 (Oberwolfach, 1993), London Math. Soc. Lecture Note Ser. 226 (Cambridge University Press, Cambridge, 1995), pp. 101–146.CrossRefGoogle Scholar
[10]Kaliszewski, S., Quigg, J. and Raeburn, I., ‘Duality of restriction and induction for C *-coactions, Trans. Amer. Math. Soc. 349 (1997), 2085–2113.CrossRefGoogle Scholar
[11]Lance, E.C., Hilbert C*-modules, London Math. Soc. Lecture Note Ser. 210 (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
[12]Mendelson, E., Introduction to mathematical logic (Van Nostrand, Princeton, NJ, 1964).Google Scholar
[13]Muhly, P. and Solel, B., ‘Tensor algebras over C *-correspondences: representations, dilations, and C *-envelopes, J. Funct. Anal. 158 (1998), 389–457.CrossRefGoogle Scholar
[14]Pimsner, M.V., ‘A class of C *-algebras generalizing both Cuntz-Krieger algebras and crossed products by Z.’, in Free Probability Theory 12, (Voiculescu, D.-V., Editor), Fields Inst. Commun. (American Mathematical Society, Providence, RI, 1997), pp. 189–212.Google Scholar
[15]Raeburn, I., ‘On crossed products and Takai duality, Proc. Edinburgh Math. Soc. 31 (1988), 321–330.CrossRefGoogle Scholar
[16]Raeburn, I. and Williams, D.P., Morita equivalence and continuous-trace C *-algebras (American Mathematical Society, Providence, RI, 1998).CrossRefGoogle Scholar
[17]Takesaki, M., ‘Covariant representations of C *-algebras and their locally compact automorphism groups, Acta Math. 119 (1967), 273–303.CrossRefGoogle Scholar