| dc.description.abstract | From the standpoint of non-commutative probability, we investigate operators over Fock spaces whose behavior on one level depends only on two of its neighbors. This behavior can be interpreted using the combinatorics of lattice paths and non-crossing partitions. Our first objective is to generalize (via a common framework) the results of Anshelevich (from 2007), Lenczewski & Sałapata (from 2008), and Bo˙zejko & Lytvynov (from 2009), whose constructions exhibited this behavior. We extend a number of results from these papers to our more general setting. These include the quadratic relation satisfied by the generating function for (a variant of) the free cumulants, the re-solvent form of the generating function for the Wick polynomials, and classification results for the case when the vacuum state on the operator algebra is tracial. We are able to handle the generating functions in infinitely many variables by considering their matrix-valued versions. Finally, we provide norm estimates guaranteeing that these generating functions are represented by bounded operators.
Our second objective is to focus on a specific class of examples within our framework, which generalizes the free multinomial example from Anshelevich’s work. Moreover, their distributions over the Fock space have convolution-power relations with those of the underlying elements of the originating space, seen through the viewpoint of free, Boolean, and (by considering representations with a similar construction) conditionally free cumulants. Moreover, we study the von Neumann algebras generated by these operators given various originating non-commuting probability spaces. In addition, we include the relevant background on free probability, operator algebras, and combinatorics prior to discussing these constructions and results. | |
