## PD Dr. Yana Kinderknecht

Prof. Dr. Moritz Weber

## 22nd Internetseminar on Ergodic Theorems

(Winter term 2018/2019)### News

See you in Wuppertal, 10-16 June!See below for Project 4: Mixing dynamical systems and their probabilistic interpretation.

See below for Project 13: Noncommutative ergodic theorems.

### Time and Place

Thursday, 14:30-16:00, SR9 (room 319, building E2 4)(except for 17 January 2019)

### Contents

The script will be provided here.The motivation from physics for ergodic theory is to understand the motion of a gaz in a box over the time.

Mathematically, this amounts to studying a certain linear operator on a vector space, the Koopman operator

and we may then employ techniques from operator theory. In the 22nd Internetseminar we will get to know

several ergodic theorems.

In order to get an impression of what ergodic theory is about, take a look here.

Announcement of the seminar

### About the Internetseminar

The Internetseminar on Functional Analysis was founded by the functional analysis group in Tübingen in 1997and is held since then every year. Teams from all over the world are participating (last year, 146 universities

from 40 countries were involved).

This year, lecture notes will be provided electronically every week, by Tanja Eisner (Leipzig) and

Balint Farkas (Wuppertal). We read and discuss these notes together in our local seminar. There will be an

online exchange with participants from other universities, so we will be part of a virtual seminar with many

other places. See here or also here.

The seminar is held in English. The Internetseminar aims at students on a higher level (second or third year

Bachelor's students, Master's students or PhD students) having some basic background in functional analysis.

For students who are interested, there will also be a phase 2 of the seminar from March to June 2019, where

small and internationally mixed groups work on a specific research project. The results are then presented on

a workshop in June, where all participants of the participating institutions come together.

The project from the second phase can be taken as a basis for a Bachelor's or Master's thesis.

See also here for the past 21nd Internetseminar on Functional Calculus.

### Project 4: Mixing dynamical systems and their probabilistic interpretation

(coordinator: Yana Kinderknecht)

We will study strong and weak mixing properties of a measure-preserving system; their different characterizations, their interplay with each other and with the ergodicity property. We will consider particular examples of mixing dynamical systems: Bernoulli shifts and Markov shifts (i.e., Markov chains with discrete time) with a finite state space. Note, that Markov chains are memoryless stochastic processes which have many applications in modelling of different real-life phenomena. It turns out that Bernoulli shifts are always strongly mixing. Whereas, strong mixing property of a Markov shift/chain is equivalent to its irreducibility and aperiodicity, what allows to establish asymptotics for the long-term behaviour of the process: the so-called Steady-State Theorem.
Main literature: [EFHN] Tanja Eisner, Balint Farkas, Markus Haase, Rainer Nagel, Operator Theoretic Aspects of Ergodic Theory. Springer, 2015. [W] Peter Walters, An Introduction to Ergodic Theory. Springer, 1982. [H] Olle Häggström, Finite Markov Chains and Algorithmic Applications. Cambridge University Press, 2008. Additional literature: [E] Tanja Eisner, Stability of Operators and Operator Semigroups. Operator Theory, Advances and Applications, vol. 209. Birkhäuser, 2010. [CD] Matthew A. Carlton, Jay L. Devore, Probability with Applications in Engineering, Science, and Technology. Springer, 2014. |

### Project 13: Noncommutative ergodic theorems

(coordinator: Moritz Weber)

The ergodic theorems presented in the internetseminar are certain statements about convergence
in measurable or topological spaces. Since about 80 years, also noncommutative measurable
and topological spaces are being studied: namely von Neumann algebras and C*-algebras.
They form the basis for many other concepts in mathematics involving noncommutativity such
as quantum physics, (compact) quantum groups and Connes’ noncommutative geometry, with
links to quantum information theory, just to name a few.In this project we will first make ourselves acquainted with the basic interpretation of von Neu- mann algebras resp. C*-algebras as noncommutative measurable resp. topological spaces. This
is based on Gelfand-Naimark’s Theorem. We will learn how statements
about spaces and measures are generally transformed into statements on von Neumann algebras
and states.We will then turn to noncommutative versions of some of the ergodic theorems presented in the internetseminar. Our focus is on Lance’s article from 1976 and possibly other results mentioned therein. Upon the interests of the participants of this project, we may also take into account more modern articles and approaches such as the one by Junge and Xu (2007), those by H. Huang (2016/2017), for instance his Quantum Szemeredi Theorem, or the Rokhlin dimension/property in the theory of C*-algebras and its link to dynamical systems of
C*-algebras. However, this is optional.See also the project description. Schedule for project 13:25 March - 7 April: Reading on von Neumann algebras, for instance in Functional analysis II script with the aim of understanding why commutative von Neumann algebras correspond to measurable spaces, and why noncommutative von Neumann algebras thus correspond to noncommutative measurable spaces.8 - 21 April: Understand Lance's reformulation of Birkhoff's theorem in Lance's article, page 1 and read Section 2 of Lance.22 April - 5 May: Read Section 3 and 4 of Lance.6 - 19 May: Read Section 5 and the proof of Lance Thm 5.7.20 May - 2 June: Distribute the parts of the presentation for Wuppertal and work on it.Optional: Include some reading on further noncommutative ergodic theorems like the ones in the notes by Xu and Junge, or by Huang, or by others. 2 - 9 June: Send the preliminary version of the slides/notes for the presentation to all of us and coordinate the composition of the joint presentation.10 - 16 June: Give a nice presentation at the workshop in Wuppertal!Our slot: Saturday, 15 June 2019, 14:00-16:00 (120 min.) Talk (1) [Marcell] - Von Neumann algebras as noncommutative measurable spaces Talk (2) [Johannes] - Reformulation of Birkhoff's theorem in terms of abelian von Neumann algebras Talk (3) [Mohamad] - Lance's Thm 5.7 Talk (4) [Ruxi] - Other noncommutative ergodic theorems |

### Literatur for the 22nd Internetseminar

- Online script of the course
- P. Walters,
*An introduction to ergodic theory*, Graduate Texts in Mathematics 79, Springer, 1982. - T. Eisner, B. Farkas, M. Haase, R. Nagel,
*Operator theoretic aspects of ergodic theory*, Graduate Texts

in Mathematics 272, Springer, 2015.

Last update: 29 May 2019 Moritz Weber | Impressum |