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Some irreducible free group representations in which a linear combination of the generators has an eigenvalue

Part of: Locally compact groups and their algebras Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

William L. Paschke
William L. Paschke
Affiliation:
Department of Mathematics, University of Kansas, 405 Snow Hall, Lawrence, KS 66045-2142, USA e-mail: [email protected]
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Abstract

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We construct irreducible unitary representations of a finitely generated free group which are weakly contained in the left regular representation and in which a given linear combination of the generators has an eigenvalue. When the eigenvalue is specified, we conjecture that there is only one such representation. The representation we have found is described explicitly (modulo inversion of a certain rational map on Euclidean space) in terms of a positive definite function, and also by means of a quasi-invariant probability measure on the combinatorial boundary of the group.

MSC classification

Secondary: 46L54: Free probability and free operator algebras 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 2 , April 2002 , pp. 257 - 286
Copyright
Copyright © Australian Mathematical Society 2002

References

[FTS]Figà-Talamanca, A. and Steger, T., ‘Harmonic analysis for anisotropic random walks on homogeneous trees’, Mem. Amer. Math. Soc. 110 (1994), 531.Google Scholar
[G]Greenberg, M., Lectures on algebraic topology (Benjamin, New York, 1966).Google Scholar
[H]Haagerup, U., ‘An example of a non nuclear C*-algebra which has the metric approximation property’, Invent. Math. 50 (1979), 279–293.CrossRefGoogle Scholar
[KS1]Kuhn, G. and Steger, T., ‘More irreducible boundary representations’, Duke Math. J. 82 (1996), 381–436.CrossRefGoogle Scholar
[KS2]Kuhn, G., ‘Monotony of certain free group representations’, J. Funct. Analysis 179 (2001), 1–17.CrossRefGoogle Scholar
[KS3]Kuhn, G., ‘Paschke's conjecture for endpoint anisotropic series representations of the free group’, preprint.Google Scholar
[L]Linnell, P. A., ‘Division rings and group von Neumann algebras’, Forum Math. 5 (1993), 561–576.CrossRefGoogle Scholar
[P]Paschke, W. L., ‘Pure eigenstates for the sum of generators of the free group’. Pacific J. Math. 197 (2001), 151–171; arXiv:math.OA/9906158.CrossRefGoogle Scholar
[PV]Pimsner, M. and Voiculescu, D., ‘K-groups of reduced crossed products by free groups’, J. Operator Theory 8 (1982), 131–156.Google Scholar
[Spa]Spanier, E. H., Algebraic topology (McGraw-Hill, New York, 1966).Google Scholar
[Spi]Spielberg, J., ‘Free-product groups, Cuntz-Krieger algebras, and covariant maps’, Internat. J. Math. 2 (1991), 457–476.CrossRefGoogle Scholar