We compute the moments and free cumulants of the measure , where denotes the free Poisson law with parameter t > 0. We also compute free cumulants of the symmetrization of . Finally, we introduce the free symmetrization of a probability measure on ℝ and provide some examples.
Recently, Bercovici has introduced multiplicative convolutions based on Muraki's monotone independence and shown that these convolution of probability measures correspond to the composition of some function of their Cauchy transforms. We provide a new proof of this fact based on the combinatorics of moments. We also give a new characterisation of the probability measures that can be embedded into continuous monotone convolution semigroups of probability measures on the unit circle and briefly discuss...
We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in...
The paper the title refers to is that in Proceedings of the Edinburgh Mathematical Society, 40 (1997), 367-374. Taking it as an excuse we intend to realize a twofold purpose: 1° to atomize that important result showing by the way connections which are out of favour, 2° to rectify a tiny piece of history. The objective 1° is going to be achieved by adopting means adequate to goals; it is of great gravity and this is just Mathematics. The other, 2°, comes...