# KMS States and Non-Commutative Geometry

## Topic

This meeting is an activity of the CRG on Operator Algebras and Non-Commutative Geometry.

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(Updated April 23, 2009 - Please note the dates have been changed)
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## Details

There have recently been new results on extending various aspects of the theory of KMS states from its original context of von Neumann algebras to C*-algebras and understanding the more complex structure that arises in the C*-setting (see for example, Laca-Neshveyev, arXiv:math/0304435 â€˜KMS states of quasi-free dynamics on Pimsner algebrasâ€™). KMS states lead naturally to type III von Neumann algebras. Noncommutative Geometry has recently been proposed to have application to situations where type III algebras arise. There are currently two points of view, that of Alain Connes and Henri Moscovici (arXiv:math/0609703 â€˜Type III and spectral triplesâ€™), and that of Carey-Phillips-Rennie (arXiv:0801.4605 â€˜Twisted cyclic theory and an index theory for the gauge invariant KMS state on Cuntz algebrasâ€™). These developments seem to be relevant to Bost-Connes systems, the Connes-Marcolli program in number theory, proposed applications of semifinite noncommutative geometry to invariants of C*-algebras, Hopf algebras and quantum groups, and KMS theory in NCG generally (see for example Laca-Larsen-Neshveyev, arXiv:0710.3452 â€˜On Bost-Connes type systems for number fieldsâ€™ arXiv:math/0609316 â€˜Hecke algebras of semidirect products and the finite part of the Connes-Marcolli C*-algebraâ€™).

These developments represent a very broad range of topics, some quite old and now possibly unfamiliar to many younger researchers, some very recent and as yet not widely known. Thus there is a need for a topical workshop exploring aspects of these recent developments and setting the scene for future research. There was a very successful meeting at the Hausdorff Institute for Mathematics in Bonn in July, 2008 on related topics that made very clear the great potential for rapid progress. The workshop itself is intended for about 40 people, including a number of graduate students and post-doctoral fellows.

## Additional Information

The workshop will be preceded by four days of lectures in the format of three minicourses. These will cover the key background material and will be aimed at preparing graduate students and young researchers for the workshop. We intend to have post-lecture discussion sessions and to distribute notes on some of the material.

We will have the following lecture streams:

- Four lectures on Noncommutative Geometry (Adam Rennie).
- Two introductory lectures on Spectral Flow (Alan Carey) and two lectures on von Neumann Algebras and Breuer Theory (John Phillips).
- Four lectures on C*-dynamical systems and KMS states, with examples from Cuntz-Pimsner algebras and semigroup crossed products (Marcelo Laca).

Participants: TBA

**Funding may be available to students and post-docs, please contact the organizers directly in this regard**

Accommodations: Accommodations have been blocked for participants and speakers. Individuals are responsible for securing their own accommodations. There are 25 dorm-type rooms available, and cluster houses. Each cluster house has 4 bedrooms and is completely furnished. When booking your accommodations please quote title of event "NCG & KMS States". These are on first-come first-serve basis. Please see UVic visitor's accommodations rates: http://housing.uvic.ca/visitor/visitoraccom.php We also have 10 guest suites blocked at the Executive House Hotel downtown Victoria. They can be contacted at 1-800-663-7001, http://executivehouse.com

Campus Map: http://www.uvic.ca/home/about/campus-info/maps/

Activities in Victoria: http://www.tourismvictoria.com

Evans, David (Cardiff)

Exel, Ruy (Florianopolis)

Khalkhali, Masoud (Western Ontario)

Kumjian, Alex (Reno)

Lesch, Matthias (Bonn)

Nest, Ryszard (Copenhagen)

Raeburn, Iain (Wollongong)

Renault, Jean (Orleans)

Rennie, Adam (ANU)

Sukochev, Fyodor (Adelaide)

Watatani, Yasuo (Kyushu)

**Scientific, Conference**

**June 29 â€“ July 10, 2009**

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