KU Leuven Department of Mathematical Sciences


Operator Algebra working seminar
(Monday 14-16h, seminar room B.02.18)

NOTE: In order to comply with the new measures implemented in response to the corona virus, the working seminar is in its physical form suspended until further notice.

For the benefit of the speakers, here is a clarification on the structure. The talk can start shortly after 14:05 after everyone has gathered. If not giving a 60-minute talk, please plan for a break of approximately 10-15 minutes near the middle, and try to not substantially exceed 90 minutes in total.

Possible topics to choose from: (in progress)

You can find below a selection of topics (along with descriptions when you click on them) that are suitable for holding individual or mini-series of presentations in the seminar. For many of them, two (or more) people can/should prepare the topic together and give several talks to reach a desirable level of detail. Suggestions for other topics are welcome and can be added to the list at any time.

Enveloping von Neumann algebras (taken)

Introduce the enveloping von Neumann algebra of a C*-algebra. Show that it is isometrically isomorphic to the double dual of the C*-algebra, and identify the weak-* topology at the level of the enveloping von Neumann algebra. Present a proof, using the Hahn-Banach separation theorem, that every ideal in a C*-algebra has quasi-central approximate units. One could also prove the equivariant version of this statement due to Kasparov. (A possible source for preparation can be Blackadar's book.)

An introduction to completely positive maps (taken)

Broadly speaking, cover the basics of the theory of completely positive maps between C*-algebras. Show Stinespring's theorem, which asserts that completely positive maps can be seen as compressions of *-homomorphisms. Further possibilities that can be cherry-picked: Explain injectivity for operator algebras. Discuss the minimal/maximal tensor product and study its compatibility with respect to completely positive maps. Get into a discussion about nuclearity/exactness, in particular the completely positive approximation property. Prove the Choi-Effros theorem. (The canonical source of preparation is Brown-Ozawa's book.)

Connes' theorem (open)

Provide a detailed exposition on Connes' famous result in his Annals paper that every injective factor of type II_1 is hyperfinite.

Tomita-Takesaki theory (taken)

Introduce Tomita-Takesaki theory, showing in reasonable detail why every von Neumann algebra with distinguished faithful normal state (or: semi-finite weight) carries a canonical time evolution. For two different choices of states, construct the Connes cocycle, culminating in the noncommutative Radon-Nikodym theorem. Introduce the finer subtype classification for factors of type III. Sketch why every (separable) factor of type III_lambda for 0 < lambda < 1 can be realized as a crossed product of a single (trace-scaling) automorphism on a type II_infty factor.

Automorphisms on injective type II factors (open)

This talk can only be given after the previous two. From the last one, motivate that in order to get a uniqueness theorem for injective factors of type III_lambda for 0 < lambda < 1, it suffices to classify certain single automorphisms on injective factors of type II. Reproduce Connes' noncommutative Rokhlin-type theorem for outer automorphisms on factors, and sketch how this leads to the classification of trace-scaling automorphisms on the injective II_infty factor.

Brown's theorem (taken)

Present a proof of Brown's theorem, which asserts that two sigma-unital C*-algebras are stably isomorphic if one of them can be included into the other as a full hereditary subalgebra.

Voiculescu's theorem (taken)

Present a proof of Voiculescu's famous theorem. This theorem states that any faithful essential representation of a separable C*-algebra on a Hilbert space approximately absorbs every other representation in a suitable sense. If time permits, hint at a few interesting applications. A detailed account of Voiculescu's theorem can be found near the beginning of Davidson's book.

C*-Cartan pairs and groupoid representations (taken)

Discuss (twisted) topological groupoids and their associated C*-algebras, giving some motivating examples along the way. Introduce Cartan subalgebras in the C*-algebraic context. Present Renault's theorem, asserting that every C*-Cartan pair comes from a twisted étale groupoid. Discuss the role of continuous orbit equivalence in this context. (There is a possibility to first present the analogous well-known theory in the context of von Neumann algebras.)

Extension theory for C*-algebras (open)

Develop the theory of extensions for C*-algebras, culminating in the Ext-group and (some of) its connections to K-theory. Cherrypick some nice applications: For example, present a computation for the K-theory of Cuntz algebras using extension theory. A possible (and more advanced) follow-up is to get into the theory of (twisted) extensions of C*-dynamical systems. In the long term, this topic could segway into the new approach to classification of finite C*-algebras.

Solidity for free group factors (open)

Show the Akemann-Ostrand property for free groups of several generators. Use this to prove that the free group factors are solid, and deduce for example that they are tensorially prime, meaning that they cannot decompose into any non-trivial tensor product of factors.

Schedule:

Date Speaker Title
20/04 Lukas Rollier Voiculescu's Theorem
(Note: This takes place electronically. An invitation to the online meeting through our videoconferencing system will be sent out to all participants.)
01/04 Ben Bouwen Brown's Theorem
(Note: This presentation takes place electronically and has an estimated length of 45 minutes. An invitation to the online meeting through our videoconferencing system will be sent out to all participants.)
23/03 Lise Wouters POSTPONED
16/03 Mateusz Wasilewski POSTPONED
09/03 Melvin Yeung C*-Cartan pairs and groupoid representations
02/03 Gabor Szabo Automorphisms that are uniformly close to the identity
24/02 ---
17/02 Baukje Debets Enveloping von Neumann algebras
14/01 Ben Bouwen / Lukas Rollier / Melvin Yeung Intermediate Master thesis presentations
17/12 Jonas Wahl Inductive limits of compact quantum groups and their characters
10/12 Tey Berendschot An introduction to completely positive maps
03/12 Mateusz Wasilewski An introduction to completely positive maps
19/11 Bram Verjans / Matthias Valvekens Tomita-Takesaki theory
12/11 Bram Verjans / Matthias Valvekens Tomita-Takesaki theory
05/11 Gabor Szabo Multiplier algebras of C*-algebras
22/10 Lise Wouters Elliott intertwining
15/10 Mateusz Wasilewski Multipliers on group C*-algebras
Academic Year 2018-2019
25/06 Bruno Braga Nonlinear geometry of operator spaces
04/06 Alessandro Vignati Everything you ever wanted to know about the Calkin algebra
28/05 Functional Analysis Master thesis seminar (13:30 to 15:30)
21/05 Baukje Debets On the classification of single automorphisms
07/05 Matthias Valvekens L^2-Betti numbers through the lens of the Lück dimension
23/04 Stuart White Classification of *-homomorphisms
09/04 Gabor Szabo An introduction to Matui-Sato's property (SI) and its applications
02/04 Alessandro Vignati An introduction to the Razak-Jacelon algebra
26/03 Stefaan Vaes Ergodicity and type of nonsingular Bernoulli actions
12/03 Mateusz Wasilewski Spectra of multipliers of full group C*-algebras
05/03 Chris Schafhauser Subalgebras of simple AF-algebras
26/02 Alessandro Vignati When an approximate truth is true: on, and beyond, Kirchberg's epsilon-test.
12/02 Thibault Pillon Irreducible affine isometric actions
05/02 Jorge Castillejos An introduction to Elliott intertwining
29/01 Matthias Valvekens Representations of rigid C*-tensor categories
22/01 Bram Verjans On the unique trace on every II_1 factor
18/12 Adrian Gonzalez Perez Fourier approximation for SLn
11/12 Mateusz Wasilewski Quantum isomorphisms of graphs and monoidal equivalence
04/12 Tobe Deprez Ozawa's class (S) for locally compact groups and unique prime factorisation
13/11 Gabor Szabo The McDuff theorem for strongly self-absorbing C*-algebras
(exceptionally in room B.02.16!)
06/11 Jorge Castillejos Classification of strict closures