Now showing 1 - 2 of 2
  • Publication
    Commutants of weighted shift directed graph operator algebras
    (American Mathematical Society, 2017-08) ; ;
    We consider non-selfadjoint operator algebras L(G, λ) generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs G. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for L(G, λ) in the case of the single vertex graph with two edges and a suitable choice of left weight function λ.
      316Scopus© Citations 5
  • Publication
    Private algebras in quantum information and infinite-dimensional complementarity
    We introduce a generalized framework for private quantum codes using von Neumann algebras and the structure of commutants. This leads naturally to a more general notion of complementary channel, which we use to establish a generalized complementarity theorem between private and correctable subalgebras that applies to both the finite and infinite-dimensional settings. Linear bosonic channels are considered and specific examples of Gaussian quantum channels are given to illustrate the new framework together with the complementarity theorem.
      385Scopus© Citations 13