## History and Aim of the course.

The theory of operator algebras introduced in the thirties by J. von Neumann was developed in close relationship with fundamental aspects of functional analysis, ergodic theory, harmonic analysis and quantum physics. More recently this field has shown many further fruitful interrelations with other areas of mathematics and mathematical physics.

The school part aims to train national and international Ph.D. students and young researchers with interdisciplinary interests in mathematics or mathematical physics in some fundamental aspects of operator algebras.

The workshop part aims to present some actual research topics in operator algebras and also in current research areas where operator algebras have been succesfully applied. E.g. in non-commutative Lp spaces, quantum information theory etc.

**Past and future of this topic in Spain**:

The present course is part of a recent effort in Spain directed to establish the topic of Operator Algebras and its various applications.

Our course pretends to be a natural continuation of the previous events and to prepare for the future thematic semester at CRM in 2011.

__Previous events:__

- The First international congress in Spain on the topic of operator algebras took place in June 2007 at CRM in Barcelona. See http://www.crm.es/Conferences/0607/Algebras/algebras.htm

- The Summer School
*Lluis Santaló*in July 2008 at the UIMP in Santander was a natural continuation of the previous congress. It attracted students and researchers from 12 countries.

See http://www.bduimp.es/view.php?idSeminario=2329

__Future events:__

- From February to July 2011 it has been approved a thematic semester at CRM in Barcelona on some topics within C*-algebras. There will be also in this period an international conference at the end of the semester. See http://www.crm.es/Activitats/Activitats/2010-2011/RPALGEBRAS/web-RPALGEBRAS/default.htm

### Bibliography:

Some recent background text books; more specific references will be distributed during the course.

[1] P. Ara, F. Lledó and F. Perera (eds.), *Proceedings of the School of Mathematics Lluis Santaló 2008: Aspects of Operator Algebras and Applications*, Contemporary Mathematics, joint edition AMS-RSME, in preparation, 2009/10.

[2] B. Blackadar, *Operator Algebras. The Theory of C*-Algebras and von Neumann Algebras*. Springer Verlag, Berlin, 2006.

[3] N.P. Brown and N. Ozawa, *C*-algebras and _nite-dimensional approximations*, American Mathematical Society, Providence, 2008.

[4] A. Connes (et al. eds), *Noncommutative Geometry*, Springer Verlag, 2004.

[5] R.S. Doran (ed.), *C*-Algebras: 1943-1993. A Fifty Year Celebration*, Contemporary Mathematics Vol. 167, American Mathematical Society, Providence, 1994.

[6] R.V. Kadison y J.R. Ringrose, *Fundamentals of the Theory of Operator Algebras. Vols.I and II*, American Mathematical Society, Providence, 1997.

[7] S. Sakai, *Operator Algebras in Dynamical Systems*, Cambridge University Press, Cambridge, 1991.

[8] V.S. Sunder, *An Invitation to von Neumann algebras*, Springer Verlag, 1987.

[9] M. Takesaki: *Theory of operator algebras. **Vols I,II and III*, Springer-Verlag, 2002, 2003.