arXiv subjects

Alistair Miller

Publications and source records attributed to Alistair Miller.

8 recordsLinked to original sources

arXiv2026

Invariant measures and traces on groupoid $\boldsymbol{\mathrm{C}^\ast}$-algebras

We provide sufficient conditions for the existence of a trace on the essential $\mathrm{C}^\ast$-algebra of a (not necessarily Hausdorff) étale groupoid $G$ which extends an invariant measure $μ$ on the unit space of $G$. In particular, it suffices for the isotropy groups of $G$ to be amenable, or for $G$ to be essentially free with respect to $μ$. We also show that $G$ is essentially free with respect to an invariant measure $μ$ if and only if $μ$ extends to a unique trace on the full $\mathrm{C}^\ast$-algebra of $G$. We work in the generality of possibly infinite measures and, accordingly, possibly unbounded traces. Moreover, whenever possible, we state our results for twisted groupoids. As an application, we show that gauge-invariant algebras of finite-state self-similar groups admit a unique tracial state.

math.OA↗

arXiv2026

Discretisation and independent resolutions of ample groupoids

We develop a general framework for understanding and computing both the groupoid homology of an ample groupoid and the topological K-theory of its reduced C*-algebra, based on two main ideas: discretisation and independent resolutions. Discretisation shows that a special class of ample groupoids we term independent groupoids are homologically and K-theoretically equivalent to discrete groupoids. We introduce the notion of a resolution by independent groupoids and provide a recipe for building a controlled independent resolution of a given ample groupoid of interest, leading to a systematic way of studying its homology and K-theory. In order to illustrate our general ideas and methods, we work out several concrete examples and applications. Garside categories provide a wide range of examples, including higher rank graphs, self-similar groups and spherical Artin-Tits groups. We also present an application to the homology of Stein's groups.

math.KT↗

arXiv2026

Functoriality of real crossed product K-theory spectral sequences with respect to group homomorphisms

Spectral sequences are a key tool for computing the K-theory of a crossed product C$^*$-algebra. However, the impact of a group homomorphism $Ω\colon G \to H$ on such a spectral sequence was unknown until quite recently, even when $G = \mathbb Z^\ell$, $H = \mathbb Z^{k}.$ Recent work [Mil25] of the fourth-named author in the complex case establishes that ABC spectral sequences are functorial with respect to group homomorphisms. In this paper, we obtain the analogous result for real K-theory and for united K-theory. Specifically, we first show that the ABC spectral sequence approximates KO$_*(G \ltimes_r A)$ with the group homology H$_p(G;KO_q(A))$ when $G$ is a torsion-free discrete group satisfying the Baum--Connes conjecture with coefficients in $A$. Then, for a homomorphism $Ω\colon G \to H$ of such groups with amenable kernel, and a real $H$-C$^*$-algebra $A$, we show moreover that the map in K-theory induced by the $*$-homomorphism $G \ltimes_r A \to H \ltimes_r A$ is approximated by the natural map in group homology.

math.OA↗

arXiv2025

Homology and K-theory for self-similar actions of groups and groupoids

Nekrashevych associated to each self-similar group action an ample groupoid and a $\mathrm{C}^\ast$-algebra. We perform complete computations of the homology of the groupoid and the K-theory of the $\mathrm{C}^\ast$-algebra for a myriad of examples, including the Grigorchuk group, the Grigorchuk--Erschler group, Gupta--Sidki groups, and self-similar actions of free abelian groups and lamplighter groups. The key development is the construction, for arbitrary self-similar group actions, of long exact sequences which compute the homology and K-theory in terms of the homology of the group and K-theory of the group $\mathrm{C}^\ast$-algebra via the transfer map and the virtual endomorphism. Results are proved more generally for self-similar groupoids. As a consequence of our results and recent results of X.~Li, we are able to show that Röver's simple group containing the Grigorchuk group and Thompson's group $V$ is rationally acyclic but has nontrivial Schur multiplier. We prove many more Röver--Nekrashevych groups of self-similar groups are rationally acyclic.

math.OA↗

arXiv2025

Self-adjoint traces on the Pedersen ideal of $\mathrm{C}^\ast$-algebras

In order to circumvent a fundamental issue when studying densely defined traces on $\mathrm{C}^\ast$-algebras -- which we refer to as the Trace Question -- we initiate a systematic study of the set $T_{\mathbb R}(A)$ of self-adjoint traces on the Pedersen ideal of $A$. The set $T_{\mathbb R}(A)$ is a topological vector space with a vector lattice structure, which in the unital setting reflects the Choquet simplex structure of the tracial states. We establish a form of Kadison duality for $T_{\mathbb R}(A)$ and compute $T_{\mathbb R}(A)$ for principal twisted étale groupoid $\mathrm{C}^\ast$-algebras. We also answer the Trace Question positively for a large class of $\mathrm{C}^\ast$-algebras.

math.OA↗

arXiv2024

Isomorphisms in K-theory from isomorphisms in groupoid homology theories

We prove that for torsion-free amenable ample groupoids, an isomorphism in groupoid homology induced by an étale correspondence yields an isomorphism in the K-theory of the associated $\mathrm{C}^\ast$-algebras. We apply this to extend X. Li's K-theory formula for left regular inverse semigroup $\mathrm{C}^\ast$-algebras. These results are obtained by developing the functoriality of the ABC spectral sequence.

math.KT↗

arXiv2024

Functors between Kasparov categories from étale groupoid correspondences

For an étale correspondence $Ω\colon G \to H$ of étale groupoids, we construct an induction functor $\mathrm{Ind}_Ω\colon \mathrm{KK}^H \to \mathrm{KK}^G$ between equivariant Kasparov categories. We introduce the crossed product of an $H$-equivariant correspondence by $Ω$, and use this to build a natural transformation $α_Ω\colon K_*( G \ltimes \mathrm{Ind}_Ω-) \Rightarrow K_*(H \ltimes -)$. When $Ω$ is proper these constructions naturally sit above an induced map in K-theory $K_*(C^*(G)) \to K_*(C^*(H))$.

math.OA↗

arXiv2024

Ample groupoid homology and étale correspondences

We show that étale correspondences between ample groupoids induce homomorphisms of homology groups. To complement this we explore the module categories of ample groupoids. We construct an induction-restriction adjunction for subgroupoids, which generates a procedure for building resolutions of arbitrary groupoid modules. These resolutions can be used to work with the Tor picture of groupoid homology, enabling explicit descriptions of the maps in homology induced by étale correspondences.

math.KT↗