Representations of Cuntz algebras, loop groups and wavelets

2000/08/15 by Palle E. T. Jorgensen
Mathematics · Physics and Astronomy · #math.FA #math-ph #math.MP #msc:46L60 #msc:47D25 #msc:42A16 #msc:43A65 #msc:46L45 #msc:42A65 #msc:41A15

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published as XIIIth International Congress on Mathematical Physics (London, 2000) (A. Fokas, A. Grigoryan, T. Kibble, and B. Zegarlinski, eds.), International Press, Boston, 2001, pp. 327--332 · 6 pages, LaTeX2e "amsproc" class; expanded version of an invited lecture given by the author at the International Congress on Mathematical Physics, July 2000 in London

arxiv created 2000/08/15 · arxiv updated 2009/11/30

Abstract

A theorem of Glimm states that representation theory of an NGCR C*-algebra is always intractable, and the Cuntz algebra O_N is a case in point. The equivalence classes of irreducible representations under unitary equivalence cannot be captured with a Borel cross section. Nonetheless, we prove here that wavelet representations correspond to equivalence classes of irreducible representations of O_N, and they are effectively labeled by elements of the loop group, i.e., the group of measurable functions A:T-->U_N(C). These representations of O_N are constructed here from an orbit picture analysis of the infinite-dimensional loop group.

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