Towards harmonic analysis on locally compact quantum groups
by
Matthias Neufang
Carleton University, Ottawa
Coauthors: Marius Junge, Mehrdad Kalantar, Zhong-Jin Ruan

Extending earlier work of U. Haagerup, Z.-J. Ruan, N. Spronk and I recently proved a completely isometric representation theorem for the completely bounded (Herz-Schur) multiplier algebra over the Fourier algebra A(G), where G is a locally compact group. This can be seen as dual to the representation of the measure algebra M(G) studied by F. Ghahramani, E. Stormer, and myself. In this talk, we present a unification of both results by extending the representation to arbitrary locally compact quantum groups (LCQGs), as obtained in joint work with M. Junge and Z.-J. Ruan. Our representation yields an interesting class of quantum channels, and enables us to express quantum group duality precisely in terms of a commutation relation for these channels. Moreover, my Ph.D. student M. Kalantar and I have shown that the quantum channels with noiseless error correction can be identified with the intrinsic group of the dual quantum group. We can thus assign, to each LCQG, a locally compact group that is an invariant for the latter, and indeed this functor also preserves compactness, discreteness and finiteness. Combining this construction with the above-mentioned commutation result, we can further assign, to each LCQG, a certain subgroup of the circle group that forms a numerical invariant and may be used towards a classification of LCQGs.

Date received: June 16, 2008