We define concepts of amenability and co-amenability for locally compact quantum groups in the sense of J. Kustermans and S. Vaes. Co-amenability of a lcqg (locally compact quantum group) is proved to be equivalent to a series of statements, all of which imply amenability of the dual lcqg. Further, it is shown that if a lcqg is amenable, then its universal dual lcqg is nuclear. We also define and study amenability and weak containment concepts for representations and corepresentations of lcqg's.
We study some aspects of the theory of non-commutative differential calculi over complex algebras, especially over the Hopf algebras associated to compact quantum groups in the sense of S.L. Woronowicz. Our principal emphasis is on the theory of twisted graded traces and their associated twisted cyclic cocycles. One of our principal results is a new method of constructing differential calculi, using twisted graded traces.1991 Mathematics Subject Classification. 46L, 81R50.
Abstract. For the q-deformation Gq, 0 < q < 1, of any simply connected simple compact Lie group G we construct an equivariant spectral triple which is an isospectral deformation of that defined by the Dirac operator D on G. Our quantum Dirac operator Dq is a unitary twist of D considered as an element of U g ⊗ Cl(g). The commutator of Dq with a regular function on Gq consists of two parts. One is a twist of a classical commutator and so is automatically bounded. The second is expressed in terms of the commutator of the associator with an extension of D. We show that in the case of the Drinfeld associator the latter commutator is also bounded.
We prove that for any non-trivial product-type action α of SU q (n) (0 < q < 1) on an ITPFI factor N , the relative commutant (N α ) ′ ∩ N is isomorphic to the algebra L ∞ (SU q (n)/T n−1 ) of bounded measurable functions on the quantum flag manifold SU q (n)/T n−1 . This is equivalent to the computation of the Poisson boundary of the dual discrete quantum group SU q (n). The proof relies on a connection between the Poisson integral and the Berezin transform. Our main technical result says that a sequence of Berezin transforms defined by a random walk on the dominant weights of SU (n) converges to the identity on the quantum flag manifold. This is a q-analogue of some known results on quantization of coadjoint orbits of Lie groups.
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