2021/08IWOTA, Session 2 (Operator algebras)The rank problem for C*-algebras   Download Slides
AbstractThe rank problem for a given C*-algebra is to describe the functions on its trace space that arise as the rank of an operator. This is a deep problem about the internal structure of C*-algebras, and a solution will have important consequences, in particular for the Toms-Winter regularity conjecture and the classification of simple, amenable C*-algebras.
I will explain how the geometry of the trace space creates difficulties in a solution of the rank problem, and how these difficulties were overcome in the recent solution of the rank problem for C*-algebras of stable rank one.
2021/08IWOTA, Session 16 (Operator Ideals and Operators on Banach spaces)Rigidity results for Lp-operator algebras   Download Slides
An Lp-operator algebra is a Banach algebra that admits an isometric representation on some Lp-space (p not 2). Given such an algebra A, we show that it contains a unique maximal sub-C*-algebra, which we call its C*-core. The C*-core is automatically abelian, and its spectrum is naturally equipped with an inverse semigroup of partial homeomorphisms. We call the associated groupoid of germs the Weyl groupoid of A.
The Weyl groupoid contains information about the internal dynamics of the algebra A, and in some some cases it is a complete invariant. For example, given a topologically free action on a compact space, the Weyl groupoid of the reduced Lp-crossed product is simply the transformation groupoid of the action. This leads to strong rigidity results for reduced groupoid algebras and reduced crossed products.
We use our results to answer a question of Phillips: The Lp-analog of the Cuntz algebra O_2 is not isomorphic to its tensor square.
(Joint work with Yemon Choi and Eusebio Gardella.)
2021/08Young mathematicians in C*-algebras (YMC*A), Münster
2021/07Workshop "Cuntz semigroups", Münster
2021/07Groups meet C*-algebras, Conference in celebration of Siegfried Echterhoff’s 60th birthday, Münster
2021/06Tokyo-Kyoto Joint Online Operator Algebra SeminarsThe generator rank of C*-algebras   Download Slides
2021/05Great Plains Operator Theory Symposium (GPOTS)The generator rank of C*-algebras   Download Slides
2020/11Noncommutative Geometry and Topology Group in Prague (NCG&T seminar), 24. November 2020The generator rank of C*-algebras   Download Slides   Video on Youtube
Abstract The generator problem asks to determine for a given C*-algebra its minimal number of generators. In particular, one wants to know if every separable, simple C*-algebra is generated by a single element. The generator problem was originally asked for von Neumann algebras, and Kadison included it as Nr. 14 of his famous list of 20 “Problems on von Neumann algebras”. The problem remains open, most notably for the reduced free group C*-algebras and the free group factors.
The generator rank is a stable quantification of the generator problem. Instead of asking if a C*-algebra is generated by k elements, the generator rank records whether the generating k-tuples are dense. It turns out that this invariant has good permanence properties and it can be computed in a number of interesting cases.
In this talk, I will first give an overview on the generator problem and then present some recent results on computations of the generator rank. Most interestingly, we will see a strong solution to the generator problem for separable, simple, classifiable C*-algebras: They are not merely singly generated but they contain a dense set of generators.
2020/11Oberseminar Free Probability, Saarland University, Saarbrücken, 11. November 2020Diffuse traces and Haar unitaries
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AbstractA Haar unitary (with respect to a given tracial state) is a unitary such that every nonzero power of the unitary and its adjoint has vanishing trace. We show that a tracial state admits a Haar unitary if and only if it is diffuse (the unique extension to a normal tracial state on the enveloping von Neumann algebra vanishes on every minimal projection), if and only if it does not dominate a tracial functional that factors through a finite-dimensional quotient.
It follows that a unital C*-algebra has no finite-dimensional representations if and only if each of its tracial states admits a Haar unitary. In particular, every tracial state on an infinite-dimensional, simple C*-algebra admits a Haar unitary.
I will sketch a proof of this result and present some applications to group C*-algebras and reduced free products.
2020/06UK Virtual Operator Algebras Seminar, 4. June 2020Cuntz semigroups
2019/11Richard Kadison and his mathematical legacy – A memorial conference, Copenhagen, Denmark, 29.-30. November 2019C*-algebras of stable rank one and their Cuntz semigroups
2019/11Workshop „Stability in Group Theory and Operator Algebras“, Copenhagen, Denmark, 25.-28. November 2019Rigidity results for Lp-operator algebras
2019/09II Barcelona Weekend on Operator Algebras and Noncommutative Algebra. In honour of Pere Ara for his 60th birthday“, CRM, Barcelona, Spain, 19.-21. September 2019
2019/19Workshop „Topology and Measure in Dynamics and Operator Algebras“, BIRS, Banff, Canada, 9.-13. September 2019Rigidity results for Lp-operator algebras
2019/08Workshop „C*-Algebras“, MFO, Oberwolfach, Germany, 12.-16. August 2019C*-algebras of stable rank one and their Cuntz semigroups (16. August 2019)
2019/03Thematic Research Program: Operator Algebras, Groups and Applications to Quantum Information“, ICMAT, Madrid, Spain, March 2019
School I (11.-15. March 2019)
Mini-Course (5 lectures): Structure and classification of amenable C*-algebras
A series of spectacular breakthroughs over the past three years led to the completion of the decades-long effort to classify simple, amenable C*-algebras by K-theory. We will discuss these results and the connections to the regularity conjecture of Toms-Winter on the structure of simple, amenable C*-algebras. We will also study the Cuntz semigroup, which is a geometric refinement of K-theory that has been used to obtain structure and classification results for non-simple C*-algebras.
2018/11Workshop "Model Theory and Operator Algebras", Banff, Canada, 25.-30. November 2018
2018/09Workshop on Analysis and Group Theory, Dresden, Germany, 6.-7. September 2018Rigidity results for group algebras
2018/0640. Nordwestdeutsches Funktionalanalysis Kolloquium (NWDFA), Wuppertal, Germany, 23.June 2018Rigidität von Gruppenalgebren
2018/06Mini-workshop on the Cuntz semigroup, Houston, USA, June 2018The category of abstract Cuntz semigroups
2018/05Conference "NCGOA 2018: C*-algebras and Dynamics", Münster, Germany, 14.-19.May 2018Infima in Cuntz semigroups and the structure of C*-algebras with stable rank one
2018/05Research visit to the University of Glasgow, UK, 9.-11. May 2018Infima in Cuntz semigroups and the structure of C*-algebras with stable rank one. (10. May 2018)
2018/04Research visit to the University of Copenhagen, Denmark, 1.-4. April 2018Ranks of operators in simple C*-algebras with stable rank one. (3. April 2018)
2017/12Conference "VI Coloquio Uruguayo de Matemática", Montevideo, Uruguay, 20.-22. December 2017Noncommutative absolute neighborhood retracts
2017/09Conference "Future Targets in the Classification Program for Amenable C*-Algebras", Banff, Canada, 3.-8. September 2017Unperforation and divisibility of Cuntz semigroups
2017/07Oberseminar C*-Algebren, Münster, Germany, July 2017Almost unperforation and almost divisibility in Cuntz semigroups. (4. July 2017)
2017/03-06IRP Operator Algebras: Dynamics and Interactions, Centre de Recerca Matemàtica (CRM), Barcelona, Spain 2017
6 week research visit, March and April 2017.
Workshop on C*-algebras and Dynamical Systems, Barcelona, 20.-24. March 2017
Barcelona Conference on C*-algebras: Structure, Classification and Dynamics, Barcelona, 19.-23. June 2017
Abstract bivariant Cuntz semigroups

Almost unperforation and almost divisibility in Cuntz semigroups
2016/11Conference "Structure and classification of C*-algebras", Warsaw, Poland, 21.-25. November 2016Abstract bivariant Cuntz semigroups (23. November 2016)
2016/08Workshop "C*-Algebras", Oberwolfach, Germany, 21.-27. August 2016
2016/07Young Mathematicians in C*-Algebras (YMC*A), Münster, Germany, 25.-29. July 2016.
2016/06Research visit to Universitat Autònoma de Barcelona (UAB Barcelona), Spain, 27. June – 1. July 2016Projections onto the essential space of a representation (28. June 2016)
2016/02-04Research visit to the Mittag-Leffler Institut, Stockholm, Sweden, 21. February – 2. April 2016
• Workshop Classification and set theory, 21.- 24. March 2016
27th Nordic Congress of Mathematicians, 16. – 20. March 2016
Workshop Classification and dynamical systems I: C*-algebras, 22.- 26. February 2016
Cuntz semigroups of ultraproducts. (22. March 2016)

Projections onto the essential space of a representation, Institut Mittag-Leffler Seminar, 1. March 2016
2016/01Winter School on Isomorphism Conjectures and Geometry of Groups, University of Münster, January 25 – 29, 2016
2015/11Conference "Noncommutative Dimension Theories", University of Hawai’i, USA, 22.-25. November, 2015
2015/09-10Research visit to Universitat Autònoma de Barcelona (UAB Barcelona), Spain, 28. September – 2. October 2015Rigidity results for group algebras. (28. September 2015)
2015/08Young Mathematicians in C*-Algebras (YMC*A), University of Copenhagen, Denmark, 17.-21. August 2015Abstract Bivariant Cuntz Semigroups. (17. August 2015)
2015/06-07International conference on C*-algebras and dynamical systems – in honor of George Elliott’s 70th birthday, Hebei Normal university, Shijiazhuang, China, June/July 2015Rigidity results for group algebras
2015/05Research visit to Universtity of Copenhagen, May 2015Rigidity results for group algebras. (Operator Algebra Seminar, 27. May 2015)
2015/04Workshop on "Structure and Classification of C*-algebras", Münster, Germany, 20-24. April 2015
2015/04Masterclass on "Groups, boundary actions and group C*-algebras", Copenhagen, Denmark, 13.-17. April 2015
2015/03Research in Pairs, Mathematisches Forschungsinstitut Oberwolfach (MFO), 9.-20. March 2015
with Ramon Antoine and Francesc Perera.
2014/10Oberseminar C*-Algebren, Münster, Germany, 21. October 2014Lp operator algebras.
2014/07Research visit to Eugene, USA, July 2014Induction functors for locally compact groups.
2014/06Canadian Operator Symposium (COSY), Toronto, Canada, 23.-27. June 2014Banach algebras generated by an invertible isometry of an Lp-space
2014/06Workshop "C*-Algebras and Dynamical Systems", Fields Institute, Toronto, Canada, 16.-20. June 2014.Structure of simple Cuntz-semirings. (Toronto, 17. June 2014)
2014/05Great Plains Operator Theory Symposium (GPOTS), Kansas State University, Manhattan, Kansas, USA, 27.-31. May 2014.Recasting the Cuntz category. (28. May 2014)
2014/04Research visit to Penn State University, State College, USA, April 2014Recasting the Cuntz category. (C*-algebra Seminar, Penn State University, 25. April 2014)
The generator problem for C*-algebras. (Noncommutative Geometry Seminar, Penn State University, 24. April 2014)
2014/04Research visit to University of Virginia, Charlottesville, USA, April 2014The generator problem for C*-algebras. (Seminar in operator theory and operator algebras. 22. April 2014)
2014/04Operator Algebras Seminar, Toronto, 1. May 2014The generator problem for C*-algebras
2014/01Research visit to the University of Waterloo, Canada, January 2014.Recasting the Cuntz category. (Analysis seminar, Waterloo, 31. January 2014)
2014/01Operator Algebras Seminar, Toronto, 23. January 2014Recasting the Cuntz category.
2014/01-06Thematic Program on Abstract Harmonic Analysis, Banach and Operator Algebras at The Fields Institute in Toronto, Canada, January-July 2014
Stay as postdoctoral fellow. Participation in the mini-courses, workshops and seminars of the program.
2013/11Miniworkshop „C*-Algebras, C*-Bundles and Group Actions“, Münster, Germany, November 2013co-organized with Snigdhayan Mahanta
2013/09School and workshop "Topics in Operator Algebras and Applications", Madrid, Spain, September 2013
2013/08Workshop "C*-Algebren", Oberwolfach, Germany, 26.-30. August 2013
2013/08Workshop „Groups, Dynamical Systems and C*-Algebras“, Münster, Germany, 20.-24. August 2013
2013/07Masterclass „Classification of non-simple purely infinite C*-algebras“, Copenhagen, Denmark, 22.-26. July 2013
2013/06Research visit to Universitat Autònoma de Barcelona (UAB Barcelona), Spain, 17.-21. June 2013Induction functors for locally compact groups. (19. June 2013)
2013/05Research visit to Eugene, USA, May 2013Induction functors for locally compact groups. (28. May 2013)
2013/05Great Plains Operator Theory Symposium (GPOTS), Berkeley, USA, 21.-25. May 2013Semiprojectivity with and without a group action. (24. May 2013)
2013/04Workshop „The structure and classification of nuclear C*-algebras“, Edinburgh, Great Britain, 15.-19. April 2013Equivariant Semiprojectivity. (16. April 2013)
2013/03School "Group C*-algebras", Sde Boker, Israel, 17.-21. March 2013
2013/03Workshop "C*-Algebras and Noncommutative Dynamics", Sde Boker, Israel, 11.-14. March 2013Equivariant Semiprojectivity. (11. March 2013)
2013/01Research visit to Tokyo, Japan, January 2013The generator problem for C*-algebras. (11. January 2013)
2012/12Oberseminar C*-Algebren, Münster, Germany, 18. December 2012The generator problem for Z-stable C*-algebras.
2012/11Workshop "C*-Algebras, Dynamics, and Classification", Oberwolfach, Germany, 29. October – 2. November 2012The generator rank for C*-algebras.
2012/09Workshop „Semiprojectivity“, Copenhagen, Denmark, 3.-7. September 2012Inductive limits of projective C*-algebras.
2012/08Masterclass "Semiprojectivity", Copenhagen, Denmark, 27.-31. August 2012
2012/07Research visit to Glasgow, Great Britain, July 2012The generator rank for C*-algebras. (18. July 2012)
2012/07Research visit to Barcelona, Spain, July 2012The generator rank for C*-algebras. (11. July 2012)
2012/05Great Plains Operator Theory Symposium (GPOTS), Houston, USA, 30. May – 3. June 2012Generators for C*-algebras.
2012/05Canadian Operator Symposium (COSY), Kingston, Canada, May 2012Inductive limits of semiprojective C*-algebras
2012/05NordForsk Network closing conference „Operator algebras and dynamical systems“, Gjógv, Faroe Islands, May 2012Generators for C*-algebras.
2012/03Masterclass „Quantum Groups“, Copenhagen, Denmark, March 2012The Jiang-Su algebra embeds into the reduced free group C*-algebras
2012/01Workshop "Set theory and C*-algebras", Palo Alto, USA, January 2012
2011/12Danish-Norwegian workshop on operator algebras, Lysebu, Oslo, Norway, December 2011The Jiang-Su algebra embeds into the reduced free group C*-algebras.
2011/11Masterclass „The nuclear dimension of C*-algebras“, Copenhagen, Denmark, November 2011Noncommutative dimension theories.
2011/11Workshop in honor of Kirchberg’s 65th birthday, Copenhagen, Denmark, November 2011
2011/10Operator algebra seminar, Copenhagen, Denmark, October 2011Inductive limits of one-dimensional NCCW-complexes. (26. October 2011)
2011/10Research visit to Trondheim, Norway, October 2011A survey on non-commutative dimension theory. (12. October 2011)
2011/09NordForsk Network Junior Workshop „Operator algebras and dynamical systems“, Copenhagen, Denmark, September 2011The generator problem for Z-stable C*-algebras
2011/09Conference "C*-Algebras and Related Topics", Kyoto, Japan, September 2011
2011/06Advanced Course „Dynamical Systems“, Barcelona, Spain, June 2011
2011/06Conference „Structure and classification of C*-Algebras“, Barcelona, Spain, June 2011Inductive Limits of Projective C*-algebras
2011/05Research visit to Nottingham, United Kingdom, May 2011Inductive limits of one-dimensional NCCW-complexes.
2011/04EU-NCG 4th Annual Meeting, Bucharest, Romania, April 2011Inductive Limits of Projective C*-algebras.
2011/04Workshop „Dynamics and C*-Algebras“, Barcelona, Spain, April 2011Inductive limits of one-dimensional NCCW-complexes.
2011Research visit to the CRM for Research program „The Cuntz semigroup and the classification of C*-algebras“, Barcelona, Spring 2011C*-algebras“, Barcelona, Spring 2011 A characterization of semiprojectivity for commutative C*-algebras.
2011/02Operator algebra seminar, Copenhagen, Denmark, February 2011Inductive Limits of Semiprojective C*-algebras. (2. February 2011)
2010/10Operator algebra seminar, Copenhagen, Denmark, October 2010A characterization of semiprojectivity for commutative C*-algebras. (13. October 2010)
2010/09Conference „NestFest“ – in Honor of Ryszard Nest’s 60th Birthday, Copenhagen, Denmark, September 2010
2010/09Workshop "Classification of amenable C*-algebras", Banff, Canada, September 2010A characterization of semiprojectivity for commutative C*-algebras.
2010/09School and Workshop "Topics in operator algebras and applications", Madrid, Spain, September 2010
2010/07Masterclass „Cuntz-Pimsner Algebras“, Copenhagen, Denmark, July 2010
2010/06EU-NCG 3rd Annual Meeting, Cardiff, Great Britain, Norway, June 2010The Chern character for low-dimensional spaces and applications.
2010/05Summerschool „C*-algebras and their interplay with dynamical systems“, Nordfjordeid, Norway, May-June 2010
2010/05Workshop „Semiprojectivity and Asymptotic Morphisms“, Copenhagen, Denmark, May 2010
2010/04Conference „UffeFest“ – in Celebration of Uffe Haagerup’s 60th Birthday and Danish-Norwegian Operator Algebra Seminar, Copenhagen, Denmark, April 2010
2009EU-NCG focused semester FS5 „Structure of Operator Algebras“, Copenhagen, Denmark, Fall 2009/2010
Masterclass „Classification of C*-algebras“, Copenhagen, Denmark, November 2009
Masterclass & Workshop „Von Neumann algebras and group actions“, Copenhagen, Denmark, January-February 2010
Torsion in the Elliott invariant and dimension theories of C*-algebras
2009 EU-NCG focused semester FS4 „K-theory, Number theory and NCG“, Münster, Germany, Spring 2009The Elliott conjecture and dimension theories of C*-algebras.
2007/06Conference „C*-Algebras and Their Invariants“, Barcelona, Spain, June 2007