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Tron Omland

Associate Professor - Several Complex Variables, Logic and Operator algebras

$Image of Tron Omland$

Email trono@math.uio.no

Mobile phone 97638997

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Visiting address Moltke Moes vei 35 Niels Henrik Abels hus 0851 Oslo

Postal address Postboks 1053 Blindern 0316 Oslo

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Academic interests

Functional analysis, operator algebras, C*-algebras, in particular C*-dynamical systems and crossed products by actions and coactions, group theory and projective representations of groups, and twisted group C*-algebras. More recent interests include algebraic number theory, combinatorics, cryptography, and quantum information.

Background

I'm a research scientist at the Norwegian National Security Authority (NSM) and have an adjunct position at the Department of Mathematics, University of Oslo. Here is my CV.

Grants

Personal post-doctoral research fellowship with the Research Council of Norway, 2015-2018, description.
Supplementary funding for the QOMBINE project with the Research Council of Norway, 2023-2026, description.

Tags: Mathematics, Operator algebras, Quantum information theory, Cryptography, USA

Haugland, Jan Kristian & Omland, Tron (2025). Shift-invariant transformations and almost liftings. Cryptography and Communications. ISSN 1936-2447. doi: 10.1007/s12095-025-00848-w. Full text in Research Archive Show summary
We investigate shift-invariant transformations, also known as rotation-symmetric vectorial Boolean functions, on n bits that are induced from Boolean functions on k bits, for k ≤ n. We consider such transformations that are not necessarily permutations, but are, in some sense, almost bijective, and study their cryptographic properties. In this context, we define an almost lifting as a Boolean function for which there is an upper bound on the number of collisions of its induced transformation that does not depend on n. We show that if a Boolean function with diameter k is an almost lifting, then the maximum number of collisions of its induced transformation is 2k−1 for any n. Moreover, we search for functions in the class of almost liftings that have good cryptographic properties and for which the non-bijectivity does not cause major security weaknesses. These functions generalize the well-known map χ used in the Keccak hash function
Aguilar, Konrad; Behera, Karina; Omland, Tron & Wu, Nicole (2025). On the Bures metric, C*-norm and quantum metric. Bulletin of the Australian Mathematical Society. ISSN 0004-9727. 112(2), p. 355–365. doi: 10.1017/S0004972724001217. Full text in Research Archive
Bedos, Eric Christophe & Omland, Tron (2023). C*-irreducibility for reduced twisted group C*-algebras. Journal of Functional Analysis. ISSN 0022-1236. 284(5), p. 109795–109826. doi: 10.1016/j.jfa.2022.109795. Full text in Research Archive Show summary
We study C*-irreducibility of inclusions of reduced twisted group C*-algebras and of reduced group C*-algebras. We characterize C*-irreducibility in the case of an inclusion arising from a normal subgroup, and exhibit many new examples of C*-irreducible inclusions.
Kaliszewski, S.; Omland, Tron; Quigg, Johnnie Currie & Turk, Jonathan (2023). Strong Pedersen rigidity for coactions of compact groups. International Journal of Mathematics. ISSN 0129-167X. 34(13). doi: 10.1142/S0129167X23500830. Full text in Research Archive Show summary
We prove a version of Pedersen’s outer conjugacy theorem for coactions of compact groups, which characterizes outer conjugate coactions of a compact group in terms of properties of the dual actions. In fact, we show that every isomorphism of a dual action comes from a unique outer conjugacy of a coaction, which in this context should be called strong Pedersen rigidity. We promote this to a category equivalence.
Enstad, Ulrik Bo Rufus; Jakobsen, Mads Sielemann; Luef, Franz & Omland, Tron (2022). Deformations and Balian-Low theorems for Gabor frames on the adeles. Advances in Mathematics. ISSN 0001-8708. 410. doi: 10.1016/j.aim.2022.108771. Full text in Research Archive
Li, Xin; Omland, Tron & Spielberg, Jack (2020). C*-algebras of right LCM one-relator monoids and Artin-Tits monoids of finite type. Communications in Mathematical Physics. ISSN 0010-3616. 381(3), p. 1263–1308. doi: 10.1007/s00220-020-03758-5. Full text in Research Archive Show summary
WestudyC*-algebrasgeneratedbyleftregularrepresentationsofrightLCM one-relator monoids and Artin–Tits monoids of finite type. We obtain structural results concerning nuclearity, ideal structure and pure infiniteness. Moreover, we compute K- theory. Based on our K-theory results, we develop a new way of computing K-theory for certain group C*-algebras and crossed products.
Bryder, Rasmus Sylvester; Ivanov, Nikolay A. & Omland, Tron (2020). C∗-simplicity of hnn extensions and groups acting on trees. Annales de l'Institut Fourier. ISSN 0373-0956. 70(4), p. 1497–1543. doi: 10.5802/AIF.3378. Full text in Research Archive Show summary
We study non-ascending HNN extensions acting on their Bass– Serre tree and characterize C∗-simplicity and the unique trace property by means of the kernel and quasi-kernels of the HNN extension in question. We also present a concrete example of an HNN extension that is a new example of a group that is not C∗-simple but does have the unique trace property. Additionally, we include certain more general results, mostly based on previous work of various authors, concerning C∗-simplicity of groups admitting extreme boundary actions, and in particular, groups acting on trees. Résumé. — Nous étudions les extensions HNN non-ascendantes agissant sur leur arbre de Bass–Serre and caractérisons la C∗-simplicité et la propriété d’unicité de la trace à l’aide du noyau et des quasi-noyaux des extensions HNN en question. Nous présentons aussi un exemple concret d’extension HNN fournissant un nouvel exemple de groupe C∗-simple mais ne possédant pas la propriété d’unicité de la trace. De plus, nous obtenons certains resultats plus généraux, la plupart se fondant sur des travaux antérieurs d’auteurs divers, concernant la C∗-simplicité de groupes admettant des actions extrêmes sur des bords, et, en particulier, de groupes agissant sur des arbres.
Omland, Tron (2019). Dynamical systems and operator algebras associated to Artin's representation of braid groups. Journal of operator theory. ISSN 0379-4024. 83(1), p. 55–72. doi: 10.7900/jot.2018jun12.2217. Full text in Research Archive
Kaliszewski, S.; Omland, Tron & Quigg, John (2019). The Pedersen Rigidity problem. Revista Colombiana de Matematicas. ISSN 0034-7426. 53(E), p. 237–277. doi: 10.15446/RECOLMA.V53NSUPL.84095. Full text in Research Archive
Omland, Tron (2019). Free nilpotent groups are C*-superrigid. Proceedings of the American Mathematical Society. ISSN 0002-9939. 148(1), p. 283–287. doi: 10.1090/proc/14678. Full text in Research Archive Show summary
The free nilpotent group Gm,n of class m and rank n is the free object on n generators in the category of nilpotent groups of class at most m. We show that Gm,n can be recovered from its reduced group C∗-algebra, in the sense that if H is any group such that C∗ r(H) is isomorphic to C∗ r(Gm,n), then H must be isomorphic to Gm,n.
Kaliszewski, S.; Omland, Tron & Quigg, John (2019). Rigidity theory for C*-dynamical systems and the "Pedersen rigidity problem", II. International Journal of Mathematics. ISSN 0129-167X. 30(8). doi: 10.1142/S0129167X19500381. Full text in Research Archive Show summary
This is a follow-up to a paper with the same title and by the same authors. In that paper, all groups were assumed to be abelian, and we are now aiming to generalize the results to nonabelian groups. The motivating point is Pedersen’s theorem, which does hold for an arbitrary locally compact group G G , saying that two actions (A,α) (A,α) and (B,β) (B,β) of G G are outer conjugate if and only if the dual coactions (A ⋊ α G, α ˆ ) (A⋊αG,α̂) and (B ⋊ β G, β ˆ ) (B⋊βG,β̂) of G G are conjugate via an isomorphism that maps the image of A A onto the image of B B (inside the multiplier algebras of the respective crossed products). We do not know of any examples of a pair of non-outer-conjugate actions such that their dual coactions are conjugate, and our interest is therefore exploring the necessity of latter condition involving the images; and we have decided to use the term “Pedersen rigid” for cases where this condition is indeed redundant. There is also a related problem, concerning the possibility of a so-called equivariant coaction having a unique generalized fixed-point algebra, that we call “fixed-point rigidity”. In particular, if the dual coaction of an action is fixed-point rigid, then the action itself is Pedersen rigid, and no example of non-fixed-point-rigid coaction is known.
Kaliszewski, S.; Omland, Tron & Quigg, John (2018). Rigidity theory for C*-dynamical systems and the "Pedersen rigidity problem". International Journal of Mathematics. ISSN 0129-167X. 29(3). doi: 10.1142/S0129167X18500167. Full text in Research Archive
Bédos, Erik & Omland, Tron (2018). On reduced twisted group C*-algebras that are simple and/or have a unique trace. Journal of Noncommutative Geometry. ISSN 1661-6952. 12(3), p. 947–996. doi: 10.4171/JNCG/295. Full text in Research Archive Show summary
On reduced twisted group C*-algebras that are simple and/or have a unique trace
Omland, Tron (2017). How many Pythagorean triples with a given inradius? Journal of Number Theory. ISSN 0022-314X. 170, p. 1–2. doi: 10.1016/j.jnt.2016.06.009. Full text in Research Archive
Kaliszewski, S.; Omland, Tron & Quigg, John (2017). Dualities for maximal coactions. Journal of the Australian Mathematical Society. ISSN 1446-7887. 102(2), p. 224–254. doi: 10.1017/S1446788716000215. Full text in Research Archive
Kaliszewski, S.; Omland, Tron & Quigg, John (2016). Destabilization. Expositiones mathematicae. ISSN 0723-0869. 34(1), p. 62–81. doi: 10.1016/j.exmath.2015.10.003. Full text in Research Archive
Kaliszewski, S.; Omland, Tron & Quigg, John (2016). Three versions of categorical crossed-product duality. New York journal of mathematics. 22, p. 293–339. Full text in Research Archive
Ivanov, Nikolay A. & Omland, Tron (2016). C*-simplicity of free products with amalgamation and radical classes of groups. Journal of Functional Analysis. ISSN 0022-1236. 272(9), p. 3712–3741. doi: 10.1016/j.jfa.2016.12.011. Full text in Research Archive
Barlak, Selçuk; Omland, Tron & Stammeier, Nicolai (2016). On the K-theory of C*-algebras arising from integral dynamics. Ergodic Theory and Dynamical Systems. ISSN 0143-3857. 38(3), p. 832–862. doi: 10.1017/etds.2016.63. Full text in Research Archive Show summary
We investigate the K -theory of unital UCT Kirchberg algebras QS arising from families S of relatively prime numbers. It is shown that K∗(QS) is the direct sum of a free abelian group and a torsion group, each of which is realized by another distinct C∗ -algebra naturally associated to S. The C∗ -algebra representing the torsion part is identified with a natural subalgebra AS of QS. For the K -theory of QS, the cardinality of S determines the free part and is also relevant for the torsion part, for which the greatest common divisor gS of {p−1:p∈S} plays a central role as well. In the case where |S|≤2 or gS=1 we obtain a complete classification for QS. Our results support the conjecture that AS coincides with ⊗p∈SOp. This would lead to a complete classification of QS, and is related to a conjecture about k -graphs. This research was first published in Ergodic Theory and Dynamical Research. © Cambridge University Press.
Bédos, Erik & Omland, Tron (2016). On twisted group C∗-algebras associated with FC-hypercentral groups and other related groups. Ergodic Theory and Dynamical Systems. ISSN 0143-3857. 36(6), p. 1743–1756. doi: 10.1017/etds.2015.9. Full text in Research Archive
Omland, Tron Ånen (2015). C*-algebras generated by projective representations of free nilpotent groups. Journal of operator theory. ISSN 0379-4024. 73(1), p. 3–25. doi: 10.7900/jot.2013mar06.2037. Full text in Research Archive
Omland, Tron Ånen (2014). Primeness and primitivity conditions for twisted group C*-algebras. Mathematica Scandinavica. ISSN 0025-5521. 114(2), p. 299–319. doi: 10.7146/math.scand.a-17113. Full text in Research Archive
Kaliszewski, S.; Omland, Tron & Quigg, John (2014). Cuntz-Li algebras from a-adic numbers. Revue Roumaine de Mathématiques Pures et Appliquées. ISSN 0035-3965. 59(3), p. 331–370. Full text in Research Archive
Omland, Tron Ånen (2013). C∗-Algebras Associated with a-adic Numbers. In Carlsen, Toke Meier; Eilers, Søren; Silvestrov, Sergei & Restorff, Gunnar (Ed.), Operator Algebra and Dynamics. Springer. ISSN 9783642394584. p. 223–238. doi: 10.1007/978-3-642-39459-1_11. Full text in Research Archive
Bedos, Erik Christopher & Omland, Tron Ånen (2012). The full group C*-algebra of the modular group is primitive. Proceedings of the American Mathematical Society. ISSN 0002-9939. 140(4), p. 1403–1411. Full text in Research Archive
Bedos, Erik Christopher & Omland, Tron Ånen (2011). Primitivity of some full group C*-algebras. Banach Journal of Mathematical Analysis. ISSN 2662-2033. 5(2), p. 44–58. Full text in Research Archive

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Omland, Tron (2025). Cryptography and national security in the quantum age. Full text in Research Archive
Omland, Tron (2025). Metrics on quantum channels from noncommutative geometry. Full text in Research Archive
Omland, Tron (2024). Shift-invariant functions and almost liftings. Full text in Research Archive
Omland, Tron (2024). Shift-invariant functions and almost liftings. Full text in Research Archive
Omland, Tron (2022). Open quantum walks. Full text in Research Archive
Omland, Tron (2022). Shift-invariant bijections. Full text in Research Archive
Omland, Tron (2021). C*-irreducibility for twisted group C*-algebras. Full text in Research Archive
Omland, Tron (2019). C*-algebras from rational dynamics. Full text in Research Archive
Omland, Tron (2019). Review of "C*-superrigidity of 2-step nilpotent groups" by Caleb Eckhardt and Sven Raum. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2018). Review of "Crossed products by Hecke pairs" by Rui Palma. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2018). Review of "An infinite C*-algebra with a dense, stably finite *-subalgebra." by Niels Jakob Laustsen and Jared T. White. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2018). C*-superrigidity for discrete groups. Full text in Research Archive
Omland, Tron (2018). Review of "On just-infiniteness of locally finite groups and their C*-algebras" by V. Belyaev, R. Grigorchuk, and P. Shumyatsky. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2018). Review of "C*-algebras associated to graphs of groups" by Nathan Brownlowe, Alexander Mundey, David Pask, Jack Spielberg, and Anne Thomas. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2017). Review of "Self-similar graphs, a unified treatment of Katsura and Nekrashevych C*-algebras" by Ruy Exel and Enrique Pardo. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2017). C*-simplicity and radical classes of groups. Full text in Research Archive
Omland, Tron (2017). C*-algebras arising from integral and rational dynamics. Full text in Research Archive
Omland, Tron (2017). C*-simplicity and radical classes of groups. Full text in Research Archive
Omland, Tron (2017). C*-simplicity for groups acting on trees. Full text in Research Archive
Omland, Tron (2017). Rigidity theory for C*-dynamical systems and the "Pedersen Rigidity Problem". Full text in Research Archive
Omland, Tron (2016). Dynamical systems and operator algebras associated to Artin's representation of braid groups II. Full text in Research Archive
Omland, Tron (2016). Review of "Imprimitivity theorems for weakly proper actions of locally compact groups" by Alcides Buss and Siegfried Echterhoff. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2016). Dynamical systems and operator algebras associated to Artin's representation of braid groups I. Full text in Research Archive
Omland, Tron (2016). C*-simplicity of free products with amalgamation and radical classes of groups. Full text in Research Archive
Omland, Tron (2016). C*-simplicity of free products with amalgamation and radical classes of groups. Full text in Research Archive
Omland, Tron (2016). C*-algebras arising from integral dynamics. Full text in Research Archive
Omland, Tron (2015). Review of "On certain multiplier projections" by Henning Petzka. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2015). Review of "Property T and amenable transformation group C*-algebras" by F. Kamalov. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2015). On the K-theory of certain Cuntz-Li algebras. Full text in Research Archive
Omland, Tron (2015). Review of "Groupoid equivalence and the associated iterated crossed product" by Jonathan Henry Brown, Geoff Goehle, and Dana P. Williams. Mathematical Reviews. ISSN 0025-5629. Full text in Research Archive
Omland, Tron (2015). C*-algebras arising from integral dynamics. Full text in Research Archive
Omland, Tron (2014). Simple reduced twisted group C*-algebras with unique trace. Full text in Research Archive
Omland, Tron (2014). On simplicity and uniqueness of trace for reduced twisted group C*-algebras. Full text in Research Archive
Omland, Tron (2013). Cuntz-Li algebras from a-adic numbers. Full text in Research Archive
Omland, Tron Ånen (2013). On the structure of certain C*-algebras arising from groups. Norges teknisk-naturvitenskapelige universitet. ISSN 9788247144268. Full text in Research Archive
Bedos, Eric Christophe & Omland, Tron (2009). The full group C*-algebra of the modular group is primitive. Universitetet i Oslo. Full text in Research Archive

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Published Feb. 16, 2016 8:10 AM - Last modified May 6, 2025 4:45 PM

Projects

Mathematics for quantum computation and many-body theory (QOMBINE)

Research groups

Operator algebras