arXiv subjects

Are Austad

Publications and source records attributed to Are Austad.

16 recordsLinked to original sources

arXiv2026

Compact Quantum Metric Spaces from Weakly Geodesic Length Functions on Hyperbolic Groups

We show that length functions on hyperbolic groups whose induced metrics are hyperbolic and weakly geodesic naturally give rise to compact quantum metric spaces, thus generalizing the result of Ozawa and Rieffel. As a result we obtain several new examples of compact quantum metric spaces, in particular arising from Green metrics associated with random walks on hyperbolic groups, and from proper cocompact isometric actions on proper geodesic hyperbolic spaces admitting a point with trivial stabilizer.

math.OA↗

arXiv2026

Amenability and comparison for \'etale groupoids with polynomial growth

We show that any second-countable locally compact Hausdorff \'etale groupoid with polynomial growth is topologically amenable. If moreover the groupoid is compactly generated with compact and metrizable unit space, it has weak $m$-comparison. Thus if the groupoid is also ample and minimal, it satisfies Matui's AH-conjecture.

math.DS↗

arXiv2026

Quantum metrics from length functions on \'etale groupoids

We show how to construct a compact quantum metric space from a proper continuous length function on an \'etale groupoid with compact unit space, where the unit space additionally has the structure of a compact metric space. Using compactly supported Fourier multipliers on the reduced groupoid $C^*$-algebra we provide a sufficient condition for verifying when we obtain a compact quantum metric space in this manner. The condition is sometimes also necessary, and is new even in the case of length functions on discrete groups. Lastly, we show that any AF groupoid with compact unit space can be equipped with a length function from which we obtain a compact quantum metric space, thereby providing a groupoid approach to understanding the quantum metric geometry of unital AF algebras.

math.OA↗

arXiv2025

Metrics on completely positive maps via noncommutative geometry

We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional $C^*$-algebraic analogue of the Choi-Jamio\l{}kowski isomorphism. Under suitable conditions, we show that the induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover, we show how to generate such metrics using constructions native to noncommutative geometry, by for example using external Kasparov products of spectral triples.

math.OA↗

arXiv2025

Quantum metrics from length functions on quantum groups

We study the quantum metric structure arising from length functions on quantum groups and show that for coamenable quantum groups of Kac type, the quantum metric information is captured by the algebra of central functions. Using this, we provide the first examples of length functions on (genuine) quantum groups which give rise to compact quantum metric spaces.

math.OA↗

arXiv2024

K-theory invariance of $L^p$-operator algebras associated with \'etale groupoids of strong subexponential growth

We introduce the notion of (strong) subexponential growth for \'etale groupoids and study its basic properties. In particular, we show that the K-groups of the associated groupoid $L^p$-operator algebras are independent of $p \in [1,\infty)$ whenever the groupoid has strong subexponential growth. Several examples are discussed. Most significantly, we apply classical tools from analytic number theory to exhibit an example of an \'etale groupoid associated with a shift of infinite type which has strong subexponential growth, but not polynomial.

math.OA↗

arXiv2024

The ideal separation property for reduced group $C^*$-algebras

We say that an inclusion of an algebra $A$ into a $C^*$-algebra $B$ has the ideal separation property if closed ideals in $B$ can be recovered by their intersection with $A$. Such inclusions have attractive properties from the point of view of harmonic analysis and noncommutative geometry. We establish several permanence properties of locally compact groups for which $L^1(G) \subseteq C^*_{\mathrm{red}}(G)$ has the ideal separation property.

math.OA↗

arXiv2023

Quantum metrics on crossed products with groups of polynomial growth

We show how to equip the crossed product between a group of polynomial growth and a compact quantum metric space with a compact quantum metric space structure. When the quantum metric on the base space arises from a spectral triple, which is compatible with the action of the group, we furthermore show that the crossed product becomes a spectral metric space. Lastly, we analyse the spectral triple at the crossed product level from the point of view of unbounded $KK$-theory and show that it arises as an internal Kasparov product of unbounded Kasparov modules.

math.OA↗

arXiv2023

Polynomial growth and property $RD_p$ for \'etale groupoids with applications to $K$-theory

We investigate property $RD_p$ for \'etale groupoids and apply it to $K$-theory of reduced groupoid $L^p$-operator algebras. In particular, under the assumption of polynomial growth, we show that the $K$-theory groups for a reduced groupoid $L^p$-operator algebra are independent of $p\in (1, \infty)$. We apply the results to coarse groupoids and graph groupoids.

math.OA↗

arXiv2023

Detecting ideals in reduced crossed product C*-algebras of topological dynamical systems

We introduce the $\ell^1$-ideal intersection property for crossed product C*-algebras. It is implied by C*-simplicity as well as C*-uniqueness. We show that topological dynamical systems of arbitrary lattices in connected Lie groups, arbitrary linear groups over the integers in a number field and arbitrary virtually polycyclic groups have the $\ell^1$-ideal intersection property. On the way, we extend previous results on C*-uniqueness of $\mathrm{L}^1$-groupoid algebras to the general twisted setting.

math.OA↗

arXiv2021

Groupoids and Hermitian Banach *-algebras

We study when the twisted groupoid Banach $*$-algebra $L^1(\mathcal{G},σ)$ is Hermitian. In particular, we prove that Hermitian groupoids satisfy the weak containment property. Furthermore, we find that for $L^1(\mathcal{G},σ)$ to be Hermitian it is sufficient that $L^1 (\mathcal{G}_σ)$ is Hermitian. Moreover, if $\mathcal{G}$ is ample, we find necessary conditions for $L^1(\mathcal{G},σ)$ to be Hermitian in terms of the fibers $\mathcal{G}^x_x$.

math.FA↗

arXiv2020

$C^*$-uniqueness Results for Groupoids

For a second-countable locally compact Hausdorff étale groupoid $\mathcal{G}$ with a continuous $2$-cocycle $σ$ we find conditions that guarantee that $\ell^1 (\mathcal{G},σ)$ has a unique $C^*$-norm.

math.OA↗

arXiv2020

Spectral invariance of $*$-representations of twisted convolution algebras with applications in Gabor analysis

We show spectral invariance for faithful $*$-representations for a class of twisted convolution algebras. More precisely, if $G$ is a locally compact group with a continuous $2$-cocycle $c$ for which the corresponding Mackey group $G_c$ is $C^*$-unique and symmetric, then the twisted convolution algebra $L^1 (G,c)$ is spectrally invariant in $\mathbb{B}(\mathcal{H})$ for any faithful $*$-representation of $L^1 (G,c)$ as bounded operators on a Hilbert space $\mathcal{H}$. As an application of this result we give a proof of the statement that if $Δ$ is a closed cocompact subgroup of the phase space of a locally compact abelian group $G'$, and if $g$ is some function in the Feichtinger algebra $S_0 (G')$ that generates a Gabor frame for $L^2 (G')$ over $Δ$, then both the canonical dual atom and the canonical tight atom associated to $g$ are also in $S_0 (G')$. We do this without the use of periodization techniques from Gabor analysis.

math.FA↗

arXiv2020

Heisenberg modules as function spaces

Let $Δ$ be a closed, cocompact subgroup of $G \times \widehat{G}$, where $G$ is a second countable, locally compact abelian group. Using localization of Hilbert $C^*$-modules, we show that the Heisenberg module $\mathcal{E}_Δ(G)$ over the twisted group $C^*$-algebra $C^*(Δ,c)$ due to Rieffel can be continuously and densely embedded into the Hilbert space $L^2(G)$. This allows us to characterize a finite set of generators for $\mathcal{E}_Δ(G)$ as exactly the generators of multi-window (continuous) Gabor frames over $Δ$, a result which was previously known only for a dense subspace of $\mathcal{E}_Δ(G)$. We show that $\mathcal{E}_Δ(G)$ as a function space satisfies two properties that make it eligible for time-frequency analysis: Its elements satisfy the fundamental identity of Gabor analysis if $Δ$ is a lattice, and their associated frame operators corresponding to $Δ$ are bounded.

math.OA↗

arXiv2019

Modulation Spaces as a Smooth Structure in Noncommutative Geometry

We demonstrate that a class of modulation spaces are examples of a smooth structure on the noncommutative 2-torus in the sense of recent developments in KK-theory. In addition, we prove that this class of modulation spaces can be represented as corners in operator linking algebras.

math.OA↗

arXiv2019

Gabor Duality Theory for Morita Equivalent $C^*$-algebras

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalent $C^*$-algebras where the equivalence bimodule is a finitely generated projective Hilbert $C^*$-module. These Hilbert $C^*$-modules are equipped with some extra structure and are called Gabor bimodules. We formulate a duality principle for standard module frames for Gabor bimodules which reduces to the well-known Gabor duality principle for twisted group $C^*$-algebras of a lattice in phase space. We lift all these results to the matrix algebra level and in the description of the module frames associated to a matrix Gabor bimodule we introduce $(n,d)$-matrix frames, which generalize superframes and multi-window frames. Density theorems for $(n,d)$-matrix frames are established, which extend the ones for multi-window and super Gabor frames. Our approach is based on the localization of a Hilbert $C^*$-module with respect to a trace.

math.OA↗