Recent results of M. Junge and Q. Xu on the ergodic properties of the averages of kernels in noncommutative -spaces are applied to the analysis of almost uniform convergence of operators induced by convolutions on compact quantum groups.
The model of generalized quons is described in an algebraic way as certain quasiparticle states with statistics determined by a commutation factor on an abelian group. Quantization is described in terms of quantum Weyl algebras. The corresponding commutation relations and scalar product are also given.
We review the notion of simple compact quantum groups and examples, and discuss the problem of construction and classification of simple compact quantum groups.
Two different models for a Hopf-von Neumann algebra of bounded functions on the quantum semigroup of all (quantum) permutations of infinitely many elements are proposed, one based on projective limits of enveloping von Neumann algebras related to finite quantum permutation groups, and the second on a universal property with respect to infinite magic unitaries.