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Discrete series of fusion algebras

Part of: Abstract harmonic analysis Selfadjoint operator algebras

Published online by Cambridge University Press:  09 April 2009

R. Srinivasan ,
V. S. Sunder  and
N. J. Wildberger
R. Srinivasan
Affiliation:
Indian Statistical Institute, R. V. College Post, Bangalore 560059, India e-mail: [email protected]
V. S. Sunder
Affiliation:
The Institute of Mathematical Sciences, Taramani Chennai 600113, India e-mail: [email protected]
N. J. Wildberger
Affiliation:
University of New South Wales, Sydney NSW 2052, Australia e-mail: [email protected]
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Abstract

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We show that the left regular representation of a countably infinite (discrete) group admits no finite-dimensional invariant subspaces. We also discuss a consequence of this fact, and the reason for our interest in this statement.

We then formally state, as a ‘conjecture’, a possible generalisation of the above statement to the context of fusion algebras. We prove the validity of this conjecture in the case of the fusion algebra arising from the dual of a compact Lie group.

We finally show, by example, that our conjecture is false as stated, and raise the question of whether there is a ‘good’ class of fusion algebras, which contains (a) the two ‘good classes’ discussed above, namely, discrete groups and compact group duals, and (b) only contains fusion algebras for which the conjecture is valid.

MSC classification

Secondary: 46L37: Subfactors and their classification 43A65: Representations of groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 3 , June 2002 , pp. 419 - 426
Copyright
Copyright © Australian Mathematical Society 2002

References

[1] Bloom, W. and Heyer, H., Harmonic analysis of probability measures on hypergroups (de Gruyter, Berlin, 1995).CrossRefGoogle Scholar
[2] Dixmier, J., C*-algebras (North-Holland, Amsterdam, 1981).Google Scholar
[3] Jones, V. F. R. and Sunder, V. S., Introduction to subfactors (Cambridge University Press, Cambridge, 1997).Google Scholar
[4] Kodiyalam, V. and Sunder, V. S., ‘Spectra of principal graphs’, Internat. J. Math. to appear.Google Scholar
[5] Obata, N. and Wildberger, N. J., ‘Generalized hypergroups and orthogonal polynomials’, Nagoya Math. J. 142 (1996), 67–93.CrossRefGoogle Scholar
[6] Popa, S., Private communication.Google Scholar
[7] Varadarajan, V. S., Lie groups, Lie algebras and their representations (Springer, New York, 1984).Google Scholar