Markov chains as Evans-Hudson diffusions in Fock space
Matrices induced by arithmetic functions, primes and groupoid actions of directed graphs
In this paper, we study groupoid actions acting on arithmetic functions. In particular, we are interested in the cases where groupoids are generated by directed graphs. By defining an injective map α from the graph groupoid G of a directed graph G to the algebra A of all arithmetic functions, we establish a corresponding subalgebra AG = C*[α(G)]︀ of A. We construct a suitable representation of AG, determined both by G and by an arbitrarily fixed prime p. And then based on this representation, we...
Measures on orthomodular vector space lattices
Monotone convolution semigroups
We study how a property of a monotone convolution semigroup changes with respect to the time parameter. Especially we focus on "time-independent properties": in the classical case, there are many properties of convolution semigroups (or Lévy processes) which are determined at an instant, and moreover, such properties are often characterized by the drift term and Lévy measure. In this paper we exhibit such properties of monotone convolution semigroups; an example is the concentration of the support...
Multiplication of free random variables and the -transform: the case of vanishing mean.
Multiplicative free square of the free Poisson measure and examples of free symmetrization
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.
Multiplicative functions on the lattice of non-crossing partitions and free convolution.
Multiplicative monotone convolutions
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...
New limit theorems related to free multiplicative convolution
Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure μ on [0,∞) with finite second moment, we find a scaling limit of as N goes to infinity. The -transform of its limit distribution can be represented by Lambert’s W-function. From this, we deduce that the limiting distribution is freely infinitely divisible, like the lognormal distribution in the classical case. We also show a similar limit theorem by replacing free...
New scaling of Itzykson–Zuber integrals
Non-Commutative Banach Function Spaces.
Non-commutative symmetric Markov semigroups.
Note on a construction of unbounded measures on a nonseparable Hilbert space quantum logic
On measure for projectors
An example of a finite set of projectors in is exhibited for which no 0-1 measure exists.
On measure for projectors. II
On a surprising relation between the Marchenko–Pastur law, rectangular and square free convolutions
In this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko–Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.
On asymptotic growth of the support of free multiplicative convolutions.
On basic concepts of non-commutative topology
On homomorphisms of projective lattices in complex Hilbert spaces
On limits of -norms of linear operators