rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Distributions and von Neumann Algebras over Fock Spaces with Depth-Two Action
  name:
    name part: Mashburn, Jacob W
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: Anshelevich, Michael
    role:
      role term: thesis advisor.
  name:
    name part: Texas A&M University
    role:
      role term: degree granting institution.
  type of resource: text
  genre: bibliography
  origin info:
    place:
      place term: txu
    date issued: 2023
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: remote
    extent: 1 online resource.
  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 & Sa¿²apata (from 2008), and BoË₉zejko & 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 Anshelevich⁰́₉s 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. The electronic version of this dissertation is accessible from https://hdl.handle.net/1969.1/198066
  note: by Jacob Mashburn.
  note: "Major Subject: Mathematics"
  note: Includes vita.
  note: Doctor of Philosophy Texas A&M University 2022 https://hdl.handle.net/1969.1/198066
  note: Includes bibliographical references.
  note: Text (Dissertation)
  subject:
    topic: Major Mathematics
  subject:
    topic: free probability
  subject:
    topic: von Neumann algebras
  subject:
    topic: Fock spaces
  identifier: https://hdl.handle.net/1969.1/198066
  record info:
    record content source: TXA
    record creation date: 240828
    record change date: 20240829173513.7
    record identifier: in00005468539
    language of cataloging:
      language term: eng