arXiv subjects

Alexander Mundey

Publications and source records attributed to Alexander Mundey.

8 recordsLinked to original sources

arXiv2025

Fibrewise compactifications and generalised limits in commutative and noncommutative topology

We introduce fibrewise compactifications in both the setting of locally compact Hausdorff spaces and continuous maps, and the parallel setting of $C^*$-algebras and nondegenerate multiplier-valued $*$-homomorphisms. In both situations, we use fibrewise compactifications to define regulated limits. In the topological setting, regulated limits extend classical inverse limits so that the resulting limit space remains locally compact; examples include the path spaces of directed graphs. In the operator-algebraic setting, regulated limits realise a direct-limit construction for multiplier-valued $*$-homomorphisms; examples include the cores of relative Cuntz-Pimsner algebras.

math.OA↗

arXiv2024

Self-similar groupoid actions on k-graphs, and invariance of K-theory for cocycle homotopies

We establish conditions under which an inclusion of finitely aligned left-cancellative small categories induces inclusions of twisted C*-algebras. We also present an example of an inclusion of finitely aligned left-cancellative monoids that does not induce a homomorphism even between (untwisted) Toeplitz algebras. We prove that the twisted C*-algebras of a jointly faithful self-similar action of a countable discrete amenable groupoid on a row-finite k-graph with no sources, with respect to homotopic cocycles, have isomorphic K-theory.

math.OA↗

arXiv2023

Morphisms of Cuntz-Pimsner algebras from completely positive maps

We introduce positive correspondences as right C*-modules with left actions given by completely positive maps. Positive correspondences form a semi-category that contains the C*-correspondence (Enchilada) category as a "retract". Kasparov's KSGNS construction provides a semi-functor from this semi-category onto the C*-correspondence category. The need for left actions by completely positive maps appears naturally when we consider morphisms between Cuntz-Pimsner algebras, and we describe classes of examples arising from projections on C*-correspondences and Fock spaces, as well as examples from conjugation by bi-Hilbertian bimodules of finite index.

math.OA↗

arXiv2023

Homology and twisted C*-algebras for self-similar actions and Zappa-Sz\'ep products

We study the categorical homology of Zappa-Sz\'ep products of small categories, which include all self-similar actions. We prove that the categorical homology coincides with the homology of a double complex, and so can be computed via a spectral sequence involving homology groups of the constituent categories. We give explicit formulae for the isomorphisms involved, and compute the homology of a class of examples that generalise odometers. We define the C*-algebras of self-similar groupoid actions on k-graphs twisted by 2-cocycles arising from this homology theory, and prove some fundamental results about their structure.

math.KT↗

arXiv2023

Splittings for C*-correspondences and strong shift equivalence

We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to C*-correspondences. We demonstrate that in-splits provide examples of strong shift equivalences of C*-correspondences. Furthermore, we provide a streamlined treatment of Muhly, Pask, and Tomforde's proof that any strong shift equivalence of regular C*-correspondences induces a (gauge-equivariant) Morita equivalence between Cuntz-Pimsner algebras. For topological graphs, we prove that in-splits induce diagonal-preserving gauge-equivariant *-isomorphisms in analogy with the results for Cuntz-Krieger algebras. Additionally, we examine the notion of out-splits for C*-correspondences.

math.OA↗

arXiv2021

A Closed Graph Theorem for hyperbolic iterated function systems

In this note we introduce a notion of a morphism between two hyperbolic iterated function systems. We prove that the graph of a morphism is the attractor of an iterated function system, giving a Closed Graph Theorem, and show how it can be used to approach the topological conjugacy problem for iterated function systems.

math.DS↗

arXiv2020

A Cuntz-Pimsner Model for the $C^*$-algebra of a Graph of Groups

We provide a Cuntz-Pimsner model for graph of groups $C^*$-algebras. This allows us to compute the $K$-theory of a range of examples and show that graph of groups $C^*$-algebras can be realised as Exel-Pardo algebras. We also make a preliminary investigation of whether the crossed product algebra of Baumslag-Solitar groups acting on the boundary of certain trees satisfies Poincaré duality in $KK$-theory. By constructing a $K$-theory duality class we compute the $K$-homology of these crossed products.

math.OA↗

arXiv2016

C*-algebras associated to graphs of groups

To a large class of graphs of groups we associate a C*-algebra universal for generators and relations. We show that this C*-algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple C*-algebras associated to generalised Baumslag-Solitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra.

math.OA↗