Are Austad

• Austad, Are; Ortega, Eduard & Palmstrøm, Mathias Sperre (2026). K-theory invariance of $L^{p}$-operator algebras associated with étale groupoids of strong subexponential growth. Documenta Mathematica. ISSN 1431-0635. doi: 10.4171/dm/1066. Full text in Research Archive Show summary

  We introduce the notion of (strong) subexponential growth for étale groupoids and study its basic properties. In particular, we show that the K-groups of the associated reduced 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 étale groupoid associated with a shift of infinite type which has strong subexponential growth, but not polynomial.

• Austad, Are & Raum, Sven (2025). DETECTING IDEALS IN REDUCED CROSSED PRODUCT C*-ALGEBRAS OF TOPOLOGICAL DYNAMICAL SYSTEMS. Journal of the Institute of Mathematics of Jussieu. ISSN 1474-7480. p. 25–25. doi: 10.1017/s147474802510145x. Full text in Research Archive Show summary

  We introduce the ℓ1-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 ℓ1-ideal intersection property. On the way, we extend previous results on C∗-uniqueness of L1-groupoid algebras to the general twisted setting.

• Austad, Are & Kyed, David (2025). Quantum metrics from length functions on quantum groups. Journal of Functional Analysis. ISSN 0022-1236. 290(4). doi: 10.1016/j.jfa.2025.111256. Full text in Research Archive Show summary

  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.

• Austad, Are & Thiel, Hannes (2025). The ideal separation property for reduced group C*-algebras. Journal of Functional Analysis. ISSN 0022-1236. 289(1). doi: 10.1016/j.jfa.2025.110904. Full text in Research Archive Show summary

  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 L1(G) ⊆ C*red(G) has the ideal separation property.

• Austad, Are; Kaad, Jens & Kyed, David (2024). Quantum metrics on crossed products with groups of polynomial growth. Transactions of the American Mathematical Society. ISSN 0002-9947. 378(3), p. 1939–1973. doi: 10.1090/tran/9340. Full text in Research Archive Show summary

  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.

• Austad, Are; Esparza, Eduardo Ortega & Palmstrøm, Mathias Sperre (2024). Polynomial growth and property RD<inf>p</inf> for étale groupoids with applications to K-theory. Journal of Noncommutative Geometry. ISSN 1661-6952. 19(2), p. 601–645. doi: 10.4171/JNCG/571. Full text in Research Archive
• Austad, Are & Esparza, Eduardo Ortega (2022). Groupoids and Hermitian Banach *-algebras. International Journal of Mathematics. ISSN 0129-167X. 33(14). doi: 10.1142/S0129167X22500902. Full text in Research Archive Show summary

  We study when the twisted groupoid Banach ∗-algebra L 1 (G, σ) is Hermitian. In particular, we prove that Hermitian groupoids satisfy the weak containment property. Furthermore, we find that for L 1 (G, σ) to be Hermitian it is sufficient that L 1 (Gσ) is Hermitian. Moreover, if G is ample, we find necessary conditions for L 1 (G, σ) to be Hermitian in terms of the fibers G x x .

• Austad, Are & Luef, Franz (2021). Modulation spaces as a smooth structure in noncommutative geometry. Banach Journal of Mathematical Analysis. ISSN 2662-2033. 15, p. 1–30. doi: 10.1007/s43037-020-00117-3. Full text in Research Archive Show summary

  We demonstrate how to construct spectral triples for twisted group C^*-algebras of lattices in phase space of a second-countable locally compact abelian group using a class of weights appearing in time–frequency analysis. This yields a way of constructing quantum C^k-structures on Heisenberg modules, and we show how to obtain such structures using Gabor analysis and certain weighted analogues of Feichtinger’s algebra. We treat the standard spectral triple for noncommutative 2-tori as a special case, and as another example we define a spectral triple on noncommutative solenoids and a quantum C^k-structure on the associated Heisenberg modules.

• Austad, Are (2021). Spectral Invariance of ∗ -Representations of Twisted Convolution Algebras with Applications in Gabor Analysis. Journal of Fourier Analysis and Applications. ISSN 1069-5869. 27(3). doi: 10.1007/s00041-021-09860-z. Full text in Research Archive Show summary

  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 Gc is C∗-unique and symmetric, then the twisted convolution algebra L1(G, c) is spectrally invariant in B(H) for any faithful ∗-representation of L1(G, c) as bounded operators on a Hilbert space H. As an application of this result we give a proof of the statement that if 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 S0(G') that generates a Gabor frame for L2(G') over , then both the canonical dual atom and the canonical tight atom associated to g are also in S0(G'). We do this without the use of periodization techniques from Gabor analysis.

• Austad, Are; Jakobsen, Mads Sielemann & Luef, Franz (2020). Gabor duality theory for Morita equivalent C*-algebras. International Journal of Mathematics. ISSN 0129-167X. 31(10). doi: 10.1142/S0129167X20500731. Full text in Research Archive Show summary

  The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group C∗-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce (n,d)-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for (n,d)-matrix Gabor frames.

• Austad, Are & Esparza, Eduardo Ortega (2020). C∗-uniqueness Results for Groupoids. International Mathematics Research Notices (IMRN). ISSN 1073-7928. doi: 10.1093/imrn/rnaa225. Full text in Research Archive Show summary

  For a 2nd-countable locally compact Hausdorff étale groupoid G with a continuous 2-cocycle σ we find conditions that guarantee that ℓ1(G,σ) has a unique C∗-norm.

• Austad, Are & Enstad, Ulrik Bo Rufus (2020). Heisenberg Modules as Function Spaces. Journal of Fourier Analysis and Applications. ISSN 1069-5869. 26(24). doi: 10.1007/s00041-020-09729-7. Full text in Research Archive Show summary

  Let \Delta 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}_{\Delta }(G) over the twisted group C^*-algebra C^*(\Delta ,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}_{\Delta }(G) as exactly the generators of multi-window (continuous) Gabor frames over \Delta , a result which was previously known only for a dense subspace of \mathcal {E}_{\Delta }(G). We show that \mathcal {E}_{\Delta }(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 \Delta is a lattice, and their associated frame operators corresponding to \Delta are bounded.

View all works in NVA

• Austad, Are (2026). Fourier multipliers and quantum metric geometry. Full text in Research Archive
• Austad, Are (2026). A new characterization of discrete groups yielding compact quantum metric spaces. Full text in Research Archive
• Austad, Are (2025). A new characterization of discrete groups yielding compact quantum metric spaces. Full text in Research Archive
• Austad, Are (2025). A new characterization of discrete groups yielding compact quantum metric spaces. Full text in Research Archive
• Austad, Are (2025). Generating hermitian algebras from noncommutative geometry. Full text in Research Archive
• Austad, Are (2025). Hermitian groupoids. Full text in Research Archive
• Austad, Are (2025). New classes of hermitian algebras. Full text in Research Archive
• Austad, Are (2024). Quantum metrics on crossed products with groups of polynomial growth. Full text in Research Archive
• Austad, Are (2024). Compact quantum metric spaces and groups with rapid decay. Full text in Research Archive
• Austad, Are (2024). The ideal separation property for dense inclusions of ∗-algebras in C* -algebras. Full text in Research Archive
• Austad, Are; Luef, Franz & Ortega, Eduard (2021). Twisted Convolution Algebras and Applications to Gabor Analysis. Norges teknisk-naturvitenskapelige universitet. ISSN 9788232669752. Full text in Research Archive Show summary

  Sammendrag: Denne avhandlingen omfatter flere aspekter ved tvistede konvolusjonsalgebraer, med spesielt fokus på problemer som oppstår i Gaboranalyse. En stor del av avhandlingen er dedikert til studiet av Hilbert C -moduler kjent som Heisenbergmoduler og hvordan disse relateres til teorien om Gaborrammer. Denne relasjonen viser sammenhengen mellom endelige Hilbert C -modulrammer og Gaborrammer. Videre omfatter avhandlingen enkelte egenskaper ved tvistede konvolusjonsalgebraer, spesielt spektralinvarians ogC -entydighet, og vi finner anvendelser for begge disse konseptene i Gaboranalyse. Spørsmålet om C -entydighet blir også bektraktet for tvistede konvolusjonsalgebraer relatert til annentellbare lokalkompakte Hausdorff étalegruppoider

View all works in NVA