skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: Fields of locally compact quantum groups: Continuity and pushouts
We prove that discrete compact quantum groups (or more generally locally compact, under additional hypotheses) with coamenable dual are continuous fields over their central closed quantum subgroups, and the same holds for free products of discrete quantum groups with coamenable dual amalgamated over a common central subgroup. Along the way we also show that free products of continuous fields of [Formula: see text]-algebras are again free via a Fell-topology characterization for [Formula: see text]-field continuity, recovering a result of Blanchard’s in a somewhat more general setting.  more » « less
Award ID(s):
2001128
PAR ID:
10337059
Author(s) / Creator(s):
Date Published:
Journal Name:
International Journal of Mathematics
Volume:
32
Issue:
09
ISSN:
0129-167X
Page Range / eLocation ID:
2150064
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, International Journal of Mathematics)
    Starting with a vertex-weighted pointed graph [Formula: see text], we form the free loop algebra [Formula: see text] defined in Hartglass–Penneys’ article on canonical [Formula: see text]-algebras associated to a planar algebra. Under mild conditions, [Formula: see text] is a non-nuclear simple [Formula: see text]-algebra with unique tracial state. There is a canonical polynomial subalgebra [Formula: see text] together with a Dirac number operator [Formula: see text] such that [Formula: see text] is a spectral triple. We prove the Haagerup-type bound of Ozawa–Rieffel to verify [Formula: see text] yields a compact quantum metric space in the sense of Rieffel. We give a weighted analog of Benjamini–Schramm convergence for vertex-weighted pointed graphs. As our [Formula: see text]-algebras are non-nuclear, we adjust the Lip-norm coming from [Formula: see text] to utilize the finite dimensional filtration of [Formula: see text]. We then prove that convergence of vertex-weighted pointed graphs leads to quantum Gromov–Hausdorff convergence of the associated adjusted compact quantum metric spaces. As an application, we apply our construction to the Guionnet–Jones–Shyakhtenko (GJS) [Formula: see text]-algebra associated to a planar algebra. We conclude that the compact quantum metric spaces coming from the GJS [Formula: see text]-algebras of many infinite families of planar algebras converge in quantum Gromov–Hausdorff distance. 
    more » « less
  2. ; ; (, Journal of Topology and Analysis)
    We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors. 
    more » « less
  3. ; ; (, International Journal of Modern Physics A)
    In this paper, we extend the non-Abelian mirror proposal of two of the authors from two-dimensional gauge theories with connected gauge groups to the case of [Formula: see text] gauge groups with discrete theta angles. We check our proposed extension by counting and comparing vacua in mirrors to the known dual two-dimensional [Formula: see text] gauge theories. The mirrors in question are Landau–Ginzburg orbifolds, and for mirrors to [Formula: see text] gauge theories, the critical loci of the mirror superpotential often intersect fixed-point loci, so that to count vacua, one must take into account the twisted sector contributions. This is a technical novelty relative to the mirrors of gauge theories with connected gauge groups, for which critical loci do not intersect fixed-point loci and so no orbifold twisted sector contributions are pertinent. The vacuum computations turn out to be a rather intricate test of the proposed mirrors, in particular as untwisted sector states in the mirror to one theory are often exchanged with twisted sector states in the mirror to the dual. In cases with nontrivial IR limits, we also check that the central charges computed from the Landau–Ginzburg mirrors match those expected for the IR SCFTs. 
    more » « less
  4. ; ; ; ; ; ; (, Journal of Mathematical Logic)
    We give examples of (i) a simple theory with a formula (with parameters) which does not fork over [Formula: see text] but has [Formula: see text]-measure 0 for every automorphism invariant Keisler measure [Formula: see text] and (ii) a definable group [Formula: see text] in a simple theory such that [Formula: see text] is not definably amenable, i.e. there is no translation invariant Keisler measure on [Formula: see text]. We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups. 
    more » « less
  5. ; ; ; ; ; (, Reviews in Mathematical Physics)
    We consider how the outputs of the Kadison transitivity theorem and Gelfand–Naimark–Segal (GNS) construction may be obtained in families when the initial data are varied. More precisely, for the Kadison transitivity theorem, we prove that for any nonzero irreducible representation [Formula: see text] of a [Formula: see text]-algebra [Formula: see text] and [Formula: see text], there exists a continuous function [Formula: see text] such that [Formula: see text] for all [Formula: see text], where [Formula: see text] is the set of pairs of [Formula: see text]-tuples [Formula: see text] such that the components of [Formula: see text] are linearly independent. Versions of this result where [Formula: see text] maps into the self-adjoint or unitary elements of [Formula: see text] are also presented. Regarding the GNS construction, we prove that given a topological [Formula: see text]-algebra fiber bundle [Formula: see text], one may construct a topological fiber bundle [Formula: see text] whose fiber over [Formula: see text] is the space of pure states of [Formula: see text] (with the norm topology), as well as bundles [Formula: see text] and [Formula: see text] whose fibers [Formula: see text] and [Formula: see text] over [Formula: see text] are the GNS Hilbert space and closed left ideal, respectively, corresponding to [Formula: see text]. When [Formula: see text] is a smooth fiber bundle, we show that [Formula: see text] and [Formula: see text] are also smooth fiber bundles; this involves proving that the group of ∗-automorphisms of a [Formula: see text]-algebra is a Banach Lie group. In service of these results, we review the topology and geometry of the pure state space. A simple non-interacting quantum spin system is provided as an example illustrating the physical meaning of some of these results. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email