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1. Continuous dependence on the initial data in the Kadison transitivity theorem and GNS construction

   https://doi.org/10.1142/S0129055X22500313  

   Spiegel, Daniel ; Moreno, Juan ; Qi, Marvin ; Hermele, Michael ; Beaudry, Agnès ; Pflaum, Markus J. ( October 2022 , 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. 

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2. Quantum Galois groups of subfactors

   https://doi.org/10.1142/S0129167X22500136  

   Bhattacharjee, Suvrajit ; Chirvasitu, Alexandru ; Goswami, Debashish ( February 2022 , International Journal of Mathematics)

   For a finite-index [Formula: see text] subfactor [Formula: see text], we prove the existence of a universal Hopf ∗-algebra (or, a discrete quantum group in the analytic language) acting on [Formula: see text] in a trace-preserving fashion and fixing [Formula: see text] pointwise. We call this Hopf ∗-algebra the quantum Galois group for the subfactor and compute it in some examples of interest, notably for arbitrary irreducible finite-index depth-two subfactors. Along the way, we prove the existence of universal acting Hopf algebras for more general structures (tensors in enriched categories), in the spirit of recent work by Agore, Gordienko and Vercruysse. 

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3. Extending group actions on metric spaces

   https://doi.org/10.1142/S1793525319500584  

   Abbott, Carolyn ; Hume, David ; Osin, Denis ( September 2020 , Journal of Topology and Analysis)

   We address the following natural extension problem for group actions: Given a group [Formula: see text], a subgroup [Formula: see text], and an action of [Formula: see text] on a metric space, when is it possible to extend it to an action of the whole group [Formula: see text] on a (possibly different) metric space? When does such an extension preserve interesting properties of the original action of [Formula: see text]? We begin by formalizing this problem and present a construction of an induced action which behaves well when [Formula: see text] is hyperbolically embedded in [Formula: see text]. Moreover, we show that induced actions can be used to characterize hyperbolically embedded subgroups. We also obtain some results for elementary amenable groups. 

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4. Weakly minimal groups with a new predicate

   https://doi.org/10.1142/S0219061320500117  

   Conant, Gabriel ; Laskowski, Michael C. ( August 2020 , Journal of Mathematical Logic)

   Fix a weakly minimal (i.e. superstable [Formula: see text]-rank [Formula: see text]) structure [Formula: see text]. Let [Formula: see text] be an expansion by constants for an elementary substructure, and let [Formula: see text] be an arbitrary subset of the universe [Formula: see text]. We show that all formulas in the expansion [Formula: see text] are equivalent to bounded formulas, and so [Formula: see text] is stable (or NIP) if and only if the [Formula: see text]-induced structure [Formula: see text] on [Formula: see text] is stable (or NIP). We then restrict to the case that [Formula: see text] is a pure abelian group with a weakly minimal theory, and [Formula: see text] is mutually algebraic (equivalently, weakly minimal with trivial forking). This setting encompasses most of the recent research on stable expansions of [Formula: see text]. Using various characterizations of mutual algebraicity, we give new examples of stable structures of the form [Formula: see text]. Most notably, we show that if [Formula: see text] is a weakly minimal additive subgroup of the algebraic numbers, [Formula: see text] is enumerated by a homogeneous linear recurrence relation with algebraic coefficients, and no repeated root of the characteristic polynomial of [Formula: see text] is a root of unity, then [Formula: see text] is superstable for any [Formula: see text]. 

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5. Metric Thickenings and Group Actions

   https://doi.org/10.1142/S1793525320500569  

   Adams, Henry ; Heim, Mark ; Peterson, Chris ( January 2021 , Journal of Topology and Analysis)

   null (Ed.)

   Let [Formula: see text] be a group acting properly and by isometries on a metric space [Formula: see text]; it follows that the quotient or orbit space [Formula: see text] is also a metric space. We study the Vietoris–Rips and Čech complexes of [Formula: see text]. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes. 

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