Email:
sven.raum@uni-potsdam.de
ORCID:
0000-0002-9188-9890
ResearcherID:
K-6725-2017
I am Sven Raum, chair of algebra at the University of Potsdam in Germany. My research focuses on operator algebras and their interaction with groups and group-like structures, such as Hecke algebras, groupoids, quantum groups and tensor categories. Important links between operator algebras and these objects are given by unitary representations and the algebras they generate, for example reduced group C*-algebras and group von Neumann algebras.
Currently, the group consists of my postdocs Sanaz Pooya and Jonathan Taylor.
I am member of the MATH+ Faculty.
Besides research mathematics, I enjoy thinking about didactics of higher education. In my free time, I like learning languages, doing sports and cooking.
I will discuss the main results and consequences of recent work with Bram Mesland. Using mostly algebraic techniques, with just a little Hilbert module analysis, we present necessary and sufficient conditions for the existence of Hermitian torsion-free connections on noncommutative one-forms, such as those arising from spectral triples. Various consequences and examples will be discussed also.
Motivated by Lusztig's basis of Hecke algebras, Graham and Lehrer introduced cellular algebras in 1996. As a generalisation of Graham and Lehrer’s cellular algebras, affine cellular algebras have been introduced by Koenig and Xi in order to treat affine versions of diagram algebras like affine Hecke algebras of type A and affine Temperley–Lieb algebras in a unifying fashion. Since then several classes of algebras, like the Khovanov-Lauda- Rouquier algebras or Kleshchev’s graded quasihereditary algebras have been shown to be affine cellular.
Roughly speaking, an affine cell ideal of an algebra A with involution * is a *-ideal J that is isomorphic as an A-bimodule to a generalized matrix ring M_{n}(B) over some commutative affine k-algebra B, whose multiplication is deformed by some "sandwich" matrix ψ, i.e. the product of two matrices x and y is defined to be x ψ y. An affine cellular algebra is then in particular a *-algebra A that admits a chain of *-ideals J_{0} < J_{1} < J_{2} < … < J_{n} =A such that each quotient J_{i}/J_{i-1} is an affine cell ideal of A/J_{i-1}. In this talk I will exhibit some ring theoretical properties of affine cellular algebras. In particular we will show that any affine cellular algebra A satisfies a polynomial identity. Furthermore, we show that A can be embedded into its asymptotic algebra if the occurring commutative affine k-algebras B_{j} are reduced and the determinants of the "sandwich" matrices are non-zero divisors. As a consequence we show that the Gelfand- Kirillov dimension of A (which measures the growth of the algebra) is less or equal to the largest Krull dimension of the algebras B_{j} and that equality holds in case all affine cell ideals are finitely generated (e.g. idempotent) or if the Krull dimension of the algebras B_{j} is less or equal to 1. Special emphasis is given to the question when an affine cell ideal is idempotent, generated by an idempotent or finitely generated.
This is joined work with Paula Carvalho, Steffen Koenig and Armin Shalile
Frames provide a robust notion of infinite spanning sequences in a Hilbert space. Dually, a Riesz sequence is a strong type of linearly independent sequence. The frame property of certain structured function systems is sometimes equivalent to the Riesz sequence property of an associated function system. There are several such so-called duality principles in applied harmonic analysis, and I aim to convey in this talk that they can all be unified by a more abstract duality principle for Morita equivalence bimodules. This is joint work with Franz Luef.
A trioid is an algebraic system consisting of a set with three binary associative operations satisfying certain axioms. Trioids are a generalization of semigroups. They play a prominent role in trialgebra theory. After defining the concept of a trioid we present examples of trioids and their relationships with such algebraic structures as Poisson algebras, Leibniz algebras, dialgebras, dimonoids, digroups and n-tuple semigroups. Then we establish independence of axioms of trioids and construct absolutely and relatively free algebras in trioid variety.
Cartan subalgebras allow C*-algebras to be represented on effective twisted étale groupoids by utilising key properties of their normaliser semigroup. Isolating these key properties, we formulate an appropriate notion of "Cartan semigroup" that allows us to represent C*-algebras on even non-effective groupoids. To each such semigroup there is again a corresponding commutative "semi-Cartan subalgebra" although it may no longer be maximal commutative and, moreover, a single semi-Cartan subalgebra may arise from various different Cartan semigroups, which yield different groupoid representations. In this talk we will outline this generalised Weyl groupoid construction and compare it to some general Fell bundle representations previously investigated by the speaker.
This is work in progress with Jonathan H. Brown, Lisa Orloff-Clark, Ying-Fen Lin and Kathryn McCormick.
Nonclosed ideals of bounded operators play a prominent role in the theory of singular traces as developed by Dixmier, Connes and many others, and the Calkin correspondence is a powerful tool that can be used to answer many questions about nonclosed ideals in this context. For general C*-algebras, a systematic study of nonclosed ideals was initiated by Pedersen in the late 1960s, but much less is known in this broader setting.
We show that a not necessarily closed ideal in a C*-algebra is semiprime (that is, an intersection of prime ideals) if and only if it is closed under roots of positive elements. Quite unexpectedly, it follows that prime and semiprime ideals in C*-algebras are automatically self-adjoint. This can be viewed as a generalization of the well-known result that closed ideals in C*-algebras are semiprime and self-adjoint.
This is joint work with Eusebio Gardella and Kan Kitamura.
Finite nuclear dimension is a regularity property of C*-algebras that have played a pivotal role in the Elliott classification program of C*-algebras. It has been a key problem in the field to verify this property for crossed product C*-algebras associated to topological and C*-dynamical systems. Previous results have mainly focused on the case of free actions. In a recent preprint (joint with Hirshberg), we show that any topological action by a finitely generated virtually nilpotent group on a finite-dimensional space gives rise to a crossed product with finite nuclear dimension. This is shown by introducing a new topological-dynamical dimension concept called the long thin covering dimension. This result can be strengthened further and applied to some allosteric (and thus non-almost-finite) actions by certain wreath product groups. Another application yields the result (joint with Eckhardt) that (twisted) group C*-algebras of virtually polycyclic groups have finite nuclear dimension.
When studying quotients of C*-algebras generated by creation and annihilation operators on analogues of Fock space, the existence of a unique smallest equivariant quotient becomes an important question in the theory. When it exists, this quotient is sometimes called the co-universal quotient. The study goes back to uniqueness theorems of Cuntz, and Cuntz and Krieger, which were extended by many authors to include several broad classes of examples.
When associating Toeplitz C*-algebras to random walks, new notions of ratio-limit space and ratio-limit boundary emerge from computing natural quotients, and the question of co-universality becomes intimately related to the geometry and the dynamics on the boundary of the random walk.
In this talk I will explain how we extended results of Woess and myself to show that there is a co-universal quotient for a large class of symmetric random walks on relatively hyperbolic groups. This allowed us to make significant progress on some questions of Woess on ratio-limits for random walks on relatively hyperbolic groups, and shed light on the general question of co-universality for Toeplitz C*-algebras arising from subproduct systems.
This talk is based on joint work with Matthieu Dussaule and Ilya Gekhtman.
We introduce t.d.l.c. groups and then study their algorithmic aspects. A useful tool is the meet groupoid, which is the natural groupoid on compact open cosets, enriched by the intersection operation. Joint work with A. Melnikov.
Renault's theorem for Cartan pairs of C*-algebras states that C*-algebras with particularly nice commutative subalgebras, called Cartan subalgebras, admit twisted groupoid models. One of the conditions that Renault required of a Cartan subalgebra was that there is a conditional expectation projecting from the ambient C*-algebra down to the subalgebra. In the groupoid setting, this corresponds to restricting functions on the groupoid to the unit space of the groupoid. In my PhD thesis, I showed that this condition can be relaxed, to allow for algebras that do not project nicely down to the subalgebra, but have conditional expectations taking values in a slightly larger algebra. Such inclusions still have twisted groupoid models, but the groupoids are no longer Hausdorff.
The Rosenberg Conjecture states that for a given Banach Algebra A, the comparison map between algebraic and topological K-theory becomes an isomorphism on finite coefficients. The new framework of condensed mathematics developed by Scholze et.al. allows one to frame this question in a different way. Condensed sets are a replacement for topological spaces that work well in their interplay with algebraic constructions. In particular, if we view our Banach algebra A as a condensed ring, we can equip the algebraic K-theory groups of A with the structure of condensed abelian groups. There exists a natural notion of completion for condensed abelian groups called solidification. The map from the K-theory of A into its solidification recovers the comparion map from algebraic to topological K-theory, allowing one to phrase the Rosenberg conjecture as a topological statement about the condensed structure of the K-theory of A.
Results from a few years ago of Kennedy and Schafhauser characterize simplicity of reduced crossed products AxG, where A is a unital C*-algebra and G is a discrete group, under an assumption which they call vanishing obstruction. However, this is a strong condition that often fails, even in cases of A being finite-dimensional and G being finite.
In joint work with Shirly Geffen, we find the correct two-way characterization of when the crossed product is simple, in the case of G being an FC-hypercentral group. This is a large class of amenable groups that, in the finitely-generated setting, is known to coincide with the set of groups which have polynomial growth. With some additional effort, we can characterize the intersection property for AxG in the non-minimal setting, for the slightly less general class of FC-groups. Finally, for minimal actions of arbitrary discrete groups on unital C*-algebras, we are able to generalize a result by Hamana for finite groups, and characterize when the crossed product AxG is prime.
All of our characterizations are initially given in terms of the dynamics of G on I(A), the injective envelope of A. This gives the most elegant characterization from a theory perspective, but I(A) is in general a very mysterious object that is hard to explicitly describe. If A is separable, our characterizations are shown to be equivalent to an intrinsic condition on the dynamics of G on A itself.
The HRT conjecture states that any finite set of time-frequency shifts of a nonzero, square-integrable function is linearly independent. While the conjecture is still open, it was settled by Linnell in the case where the time-frequency shifts belong to a discrete subgroup of R^d. In this talk I will present an elementary proof of Linnell's theorem which also extends to other settings, in particular to any coherent system arising from a projective discrete series of a simply connected, nilpotent Lie group.
The talk is based on recent joint work with Jordy Timo van Velthoven.
I will discuss some recents applications and developments surrounding the Ornstein-Weiss Rokhlin lemma involving quantitative versions of orbit equivalence and questions of tileability in topological dynamics.
In the present talk I will briefly review certain aspects of the dichotomy amenable/paradoxical in the category of metric spaces and inverse semigroups. Then I will address the question of defining a reasonable metric structure on a discrete inverse semigroup and study the properties of its uniform Roe algebra like Foelner type conditions, nuclearity or exactness.
[1] F. Lledó and D. Martínez, The uniform Roe algebra of an inverse semigroup, Journal of Mathematical Analysis and Applications 499 (2021) 124996.An exotic group C*-algebra of a group G is a C*-algebra that lies naturally in between the reduced and the universal group C*-algebra of G. The existence of an exotic C*-algebra can be viewed in some sense as a refinement of the non-amenability of G. We will discuss several approaches to construct such algebras and then will describe the explicit construction of a family of exotic group C*-algebras coming from the family of piecewise-projective groups. This is a joint work with Nicolas Monod.
Coarse groups are spaces equipped with operations that satisfy the group axioms up to uniformly bounded error. The study of coarse groups, actions and homomorphisms was only very recently initiated and leads to a number of questions and research directions. This talk is an introduction to the subject (joint work with Arielle Leitner).
The virtually abelian locally compact groups were characterized in representation theoretic terms by C. Moore in the 70s as those groups for which there is a uniform bound on the degree of their irreducible representations. For compact groups, this, in turn, provides a uniform bound on the norm of a certain family of projections appearing in their L^{p}-representation theory. We shall discuss to what extent this characterizes the virtually abelian groups among the compacts.
Divisibility properties for Cuntz semigroups play an important role in the study of both simple and non-simple C*-algebras. The prime example of this is the remaining open implication of the Toms-Winter conjecture which, under the assumption of locally finite nuclear dimension, boils down to studying if strict comparison implies almost divisibility. In this talk I will recall what these divisibility properties are, focusing specifically on those introduced by Robert and Rørdam. I will discuss how these notions give rise to natural classes of C*-algebras, and explain their connection to non-simple analogues of the Toms-Winter conjecture. I will also talk about the Global Glimm Problem, which predicts that two of these properties are equivalent. The talk is based on joint work with Hannes Thiel.
During the Corona pandemic, the Wednesday Zoom Seminar combined speakers in algebraic geometry, group theory and operator algebras.
All abstracts can be found in the SMC calender: Link.
Due to the Corona pandemic, our seminar on Harmonic Analysis, Operator Algebras and Representation Theory (pronounced "Howard") is was suspended and eventually discontinued.
All abstracts can be found in the SMC calender: Link.
Twisted group C*-algebras of acylindrically hyperbolic groups have stable rank one
ArXiv (2024)
Free actions of polynomial growth Lie groups and classifiable C*-algebras
with Ulrik Enstad and Gabriel Favre
ArXiv (2023)
Detecting ideals in reduced crossed product C*-algebras of topological dynamical systems
with Are Austad
ArXiv (2023)
A dynamical approach to non-uniform density theorems for coherent systems
with Ulrik Enstad
ArXiv (2022)
The ideal intersection property for essential groupoid C*-algebras
with Matthew Kennedy, Se-Jin Kim, Xin Li and Dan Ursu
ArXiv (2021)
On the centre of Iwahori-Hecke algebras
with Timothée Marquis
J. Algebra (2023)
and the
ArXiv version
Factorial multiparameter Hecke von Neumann algebras and representations of groups acting on right-angled buildings
with Adam Skalski
J. Math. Pure Appl. 177 (2023), 265-298
and the
ArXiv version
K-theory of right-angled Hecke C*-algebras
with Adam Skalski
Adv. Math. 407 (2022) Article No. 108559
and the
ArXiv version
Amenability, proximality and higher order syndeticity
with Matthew Kennedy and Guy Salomon
Forum of Mathematics, Sigma 10 (2022) e22, 1-28
and the
ArXiv version
An algebraic characterization of ample type I groupoids
with Gabriel Favre
Semigroup Forum 104 (2022), 58-71
and the
ArXiv version
Measure equivalence for non-unimodular groups
with Juhani Koivisto and David Kyed
Transform. Groups 26 (2021), 327-346
and the
ArXiv version
Cartan subalgebras in dimension drop algebras
with Selçuk Barlak
J. Inst. Math. Jussieu 20 (2021) No. 3, 725-755.
and the
ArXiv version
Measure equivalence and coarse equivalence for locally compact amenable groups
with Juhani Koivisto and David Kyed
Group, Geom. and Dyn. 15 (2021) No. 1, 223-267
and the
ArXiv version
C*-simplicity (after Breuillard, Haagerup, Kalantar, Kennedy and Ozawa)
Sven Raum
Séminaire Bourbaki, exposé 1158, Astérisque 422 (2020)
and the
Preprint version
Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras
with Cyril Houdayer
Comm. Math. Helv. 76 (2019) No. 1, 185-219
and the
ArXiv version
C*-simplicity of locally compact Powers groups
J. Reine Angew. Math. 748 (2019) 173-205
and the
ArXiv version
Erratum to this article
C*-superrigidity of 2-step nilpotent groups
with Caleb Eckhardt
Adv. Math. 338 (2018), 175-195 and the ArXiv version
Higher l^{2}-Betti numbers of universal quantum groups
Julien Bichon and David Kyed
Can. Math. Bull. 61 (2018), No. 2, 225-235 and the ArXiv version
Traces on reduced group C*-algebras
with Matthew Kennedy
Bull. Lond. Math. Soc. 49 (2017) No. 6, 988-990 and the ArXiv version
L^{2}-Betti numbers of rigid C*-tensor categories and discrete quantum groups
with David Kyed, Stefaan Vaes and Matthias Valvekens
Analysis & PDE 10 (2017), No. 7, 1757-1791 and the ArXiv version
On the l^{2}-Betti numbers of universal quantum groups
with David Kyed
Math. Ann. 369 (2017), No. 3-4, 957–975 and the
ArXiv version
Cocompact amenable closed subgroups: weakly inequivalent representations in the left-regular representation
Int. Math. Res. Not., (2016), No. 24, 7671-7685 and the ArXiv version
The full classification of orthogonal easy quantum groups
with Moritz Weber
Commun. Math. Phy. 341 (2016) No. 3, 751-779 and the
ArXiv
Easy quantum groups and quantum subgroups of a semi-direct product quantum group
with Moritz Weber
J. Noncommu. Geo. 9 (2015) No. 4, 1261-1293 and the ArXiv version
Asymptotic structure of free Araki-Woods factors
with Cyril Houdayer
Math. Ann. 363 (2015) No. 1-2, 237-267 and the ArXiv version
Baumslag-Solitar groups, relative profinite completions and measure equivalence rigidity
with Cyril Houdayer
J. Topol. 8 (2015), 295-313 and the ArXiv
The combinatorics of an algebraic class of easy quantum groups
with Moritz Weber
Infin. Dimens. Anal. Quantum Probab. Relat. Top. 17, No. 3 (2014) and the ArXiv version
Amalgamated free product type III factors with at most one Cartan subalgebra
with Rémi Boutonnet and Cyril Houdayer
Compositio Math. 150 (2014), 143-174 and the ArXiv version
On the classification of free Bogoljubov crossed product von Neumann algebras by the integers
Groups, Geom. and Dyn. 8 (2014) No. 4, 1207-1245 and the ArXiv version
Tensor C*-categories arising as bimodule categories of II_{1} factors
with Sébastien Falguières
Adv. Math. 237 (2013), 331-359 and the ArXiv version
Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions
with Niels Meesschaert and Stefaan Vaes
Expo. Math. 31 (2013), 274-294 and the ArXiv version
Isomorphisms and fusion rules of orthogonal free quantum groups and their complexifications
Proc. Amer. Math. Soc. 140 (2012), No. 9, 3207-3218 and the ArXiv version
Lecture notes about Creutz-Peterson's "Character rigidity for lattices and commensureators"
Written for the Arbeitsgemeinschaft "Superrigidity" at MFO, 2014
Lecture C3 and Lecture C4
A connection between easy quantum groups, varieties of groups and reflection groups
with Moritz Weber
This is an arXiv preprint from 2012 which is not intended for publication. Please cite instead the appropriate article among my remaining work with Moritz.
Categories of representations and classification of von Neumann algebras and quantum groups
My PhD thesis under the supervision of Stefaan Vaes submitted in 2013 at KU Leuven.
May 2024 |
Workshop "Point sets: Dynamics, sampling, and operator algebras" Organised at the University of Oslo |
May 2023 |
School and workshop on Noncommutative Geometry and Operator algebras Organised at the Hausdorff Centre for Mathematics in Bonn |
March 2023 |
Mini-workshop on operator algebras and noncommutative geometry Meeting of Swedish operator algebraists and noncommutative geometers at Stockholm University |
November 2022 |
Mini-workshop on operator algebras and noncommutative geometry Meeting of Swedish operator algebraists and noncommutative geometers at Chalmers/University of Gothenburg |
August 2022 |
Operator algebras and noncommutative geometry Parallel session at the Nordic Congress of Mathematics |
May 2022 |
C*-algebras and geometry of groups and semigroups Workshop at the University of Oslo |
March 2022 |
Noncommutativity in the north Workshop at Chalmers/University of Gothenburg |
January 2022 |
Mini-workshop on operator algebras and noncommutative geometry Meeting of Swedish operator algebraists and noncommutative geometers at Chalmers/University of Gothenburg |
March 2021 |
C*-algebras and geometry of groups and semigroups Online workshop |
May 2017 |
Approximate lattices Two-day reading group joint between EPFL and the University of Neuchâtel |
July 2016 |
Young Mathematicians in C*-Algebras Co-organiser responsible for funding, scientific organisation and social events |
2023/2024 Winter |
Algebra Lineare Algebra und analytische Geometrie |
2023 Summer |
Algebra and number theory Geometric group theory Lie Algebras (seminar) |
2022 Autumn | Expander graphs |
2022 Spring | Linear algebra |
2021 Autumn | Linear algebra |
2021 Spring |
Geometric group theory Complex analysis |
2020 Spring |
Introduction to operator algebras Foundations of analysis |
2019 Autumn | Combinatorics |
2019 Spring |
Foundations of analysis Ordinary differential equations |
2018 Spring |
Lie groups Homology and cohomology |
2017 Autumn | Linear algebra for engineers |
2017 Spring | Abstract harmonic analysis |
2016 Autumn | Linear algebra for engineers |
2016 Summer | Abstract harmonic analysis |
2015 Winter | Von Neumann algebras and measured group theory |
2019 - 2023 | Gabriel Favre |
2022 | C*-simplicity of discrete groups and étale groupoids |
2022 | Group C*-algebras of Heisenberg groups and C*-rigidity |
2017/18 | On groups of type I |
2016/17 | C*-simplicity and factorial group von Neumann algebras |
2011/12 | Equivalence relations with prescribed fundamental group (co-supervision) |
2010/11 | Orbit equivalence of Bernoulli actions over free groups (co-supervision) |
2009/10 | There is no universal II_{1} factor (co-supervision) |
2021/22 |
Number systems beyond the reals (teacher student) The search for orthogonal Latin squares (teacher student) Measure-theoretic Probability |
2020/21 |
Decomposition theory for zero-sum games (math-economy student) Mathematical billards (teacher student) The logic behind Kőnig's lemma Quivers, path algebras and Gabriel's theorem Which schools would take me? Truth-telling in school choice with admission probability uncertainty |
2019/20 |
Quantum computing: From Shor’s algorithm to the hidden subgroup problem The travelling salesman in Sweden (teacher student) Classical finite simple groups The spectral theorem for normal operators |
2018/19 | Wallpaper groups (teacher student) |
2017/18 | K-theory and Bott periodicity |
2016/17 | Classification of closed surfaces |
Born on 9th April 1985 in Kassel, Germany.
Since Apr 2023 | W2-Professor at the University of Potsdam, Germany |
Sep 2018 - Mar 2023 | Associate Professor at Stockholm University, Sweden |
Jun - Jul 2021 | Visiting Professor at IM PAN, Poland |
Sep - Dec 2020 | Visiting Professor at IM PAN, Poland |
2016 - 2018 | Lecturer at EPFL, Switzerland |
2015 - 2016 | Marie Curie International Outgoing Fellow at the University of Münster, Germany |
2014 - 2015 | Marie Curie International Outgoing Fellow at RIMS, Kyoto, Japan |
2013 - 2014 | Post doctoral scholar at ENS Lyon, France |
2009 - 2013 | PhD scholar at KU Leuven, Belgium |
2007 - 2009 | Student teaching assistant at the University of Münster |
Oct 2019 | Docent in Mathematics at Stockholm University, Sweden |
Sep 2009 - Jun 2013 | PhD in Mathematics at KU Leuven, Belgium |
Oct 2005 - May 2009 | Diplom Mathematik at the University of Münster, Germany |
German | Mother tongue |
English | Fluent |
French | Fluent (B2) |
Dutch | Fluent (C1) |
Swedish | Fluent |
Persian | Basic knowledge |
Japanese | Baisc knowledge (JLPT N5) |