# MA8001 C*-crossed products (spring 2010)

This course gives an introduction to C*-crossed products. Among the things we will go through are the definitions and basic properties of the universal and the reduced crossed products of a locally compact group acting on a C*-algebra, and the Pimsner-Voiculescu exact sequence. The lectures will be given in turn by Magnus Landstad, Christian Skau and Toke Meier Carlsen. We also expect the participant to contribute (more on this later).

**Time and place:** Thursdays 13.15-15.00 in seminarrom 656, Sentralbygg II.

#### Literature

- [B] Bruce Blackadar: K-theory for operator algebras, Second edition, Mathematical Sciences Research Institute Publications, 5. Cambridge University Press, Cambridge, 1998. xx+300 pp. ISBN: 0-521-63532-2.

- [T] Masamichi Takesaki: Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, 127. Operator Algebras and Non-commutative Geometry, 8. Springer-Verlag, Berlin, 2003. xxii+548 pp. ISBN: 3-540-42913-1.

- [W] Dana P. Williams: Crossed Products of C*-algebras, Mathematical Surveys and Monographs, 134. American Mathematical Society, Providence, RI, 2007. xvi+528 pp. ISBN: 978-0-8218-4242-3; 0-8218-4242-0.

#### Schedule

**Jan 28**Toke Meier Carlsen: C*-dynamical systems and covariant representations (2.1 up to and including Prop 2.7, and 2.2 up to and including Definition 2.10 [W]).**Feb 4**Toke Meier Carlsen: Covariant representations and the crossed product (the rest of 2.2 and 2.3 up to and including Prop. 2.23 of [W]).**Feb 10**Magnus Landstad: The crossed product (the rest of 2.3 of [W]).**Feb 18**Magnus Landstad: Representations of crossed products (all of 2.4 of [W] except 2.38 and 2.39).**Feb 25**Magnus Landstad: Takai duality (7.1 of [W]).**March 4**Magnus Landstad: Takai duality (7.1 of [W]).**March 11**Magnus Norling: The index map (8.3 of [B]).**March 17**Magnus Norling: The index map and Connes' Thom isomorphism (8.3, 10.2 and 10.9 of [B]).**March 24**Magnus Norling: Connes' Thom isomorphism (10.9 of [B]).**March 25**Magnus Norling: The Pimsner-Voiculescu exact sequence (10.3-10.4 of [B]).**April 14**Tron Omland: Amenable groups (Chapter XIII §4 of [T] and A2 of [W]).**April 22**Tron Omland: Amenable groups (Chapter XIII §4 of [T] and A2 of [W]).