Seminar on crossed products and Rokhlin-type properties

This is a Masters/PhD seminar offered in the Winter semester 2017-2018 at WWU Münster by Prof. Siegfried Echterhoff and Eusebio Gardella. It will take place on Thursdays at 14:00 (cum tempo), in room SR4 (fourth floor of the main Math building).

Schedule and contents:

We organized the lectures in groups of two, and we expect both lectures in each group to be prepared by both students, or at least with very close cooperation.
  • Lectures 1 and 2 (October 12 and 19): Definition of maximal and reduced crossed products and some properties (following [BO, Section 4.1]. Use also [Ech]). Definitions and discussion of full and reduced group algebras as crossed products with the complex numbers (compare with [BO, Section 2.5]). Crossed products by amenable groups ([BO, Sections 2.6 & 4.2]. Exactness of the maximal crossed product (as a functor) and exact groups ([Ech]). Speakers: Martin Stückemann and Marzieh Forough.

  • Lectures 3 and 4 (October 26 and November 2): Reduced crossed products of outer actions on simple C*-algebras are simple (Literature [Kis, Ell]). (Requires some introductory work on the double dual A** -- the enveloping von Neumann algebra of a C*-algebra.) Speakers: Fabian Felker and Siegfried Echterhoff.

  • Lectures 5 and 6 (November 9 and 16): Simplicity of reduced crossed products by topologically free minimal actions; ideal structure of reduced crossed products by essentially free actions of exact groups ([AS], [Sie] , [EL]). Speakers: Hannes Ortemeier and Frederick Groß-Bötling.

  • Lectures 7 and 8 (November 23 and 30): Definition of the Rokhlin property, the commutative case, canonical example on the UHF-algebra, some non-examples on irrational rotation algebras, Cuntz algebras, UHF-algebras (pages 5 and 6 of [Gar1]), Theorem 2.5 and Example 2.9 in [Gar2], genericity (Subsection 2.2 in [Gar2]), map on K-theory and formula for the K-theory of the fixed point algebra (Theorem 3.13 in [Izu], but there is a simpler proof), duality (Subsection 3.2 in [Izu]). Speakers: Alexander Frei and Sabrina Gemsa.

  • Lectures 9 and 10 (December 7 and 14): Study of crossed products by actions with the Rokhlin property; start with Sierakowski's application to the Rokhlin property ([Sie]), showing that all ideals in the crossed product come from invariant ideals. For structural results, the main point is to show that there is a sequence of unital completely positive maps from the algebra to the fixed point subalgebra, which leave the fixed point subalgebra fixed and are asymptotically multiplicative (Theorem 3.1 in [Gar2]; the proof gets a lot simpler for finite groups). As an application, having stable rank one, and being AF are preserved (Propositions 2.12 and 2.13 in [Gar1]). If there is enough time, other preservation results may be proved. Classification (Subsection 3.1 in [Izu]) and consequences (Corollary 2.15 in [Gar1]). Speakers: Timo Siebenand and André Schemaitat.

  • (No Seminar on December 21.)

  • Lectures 11 and 12 (January 11 and 18): Definition of the tracial Rokhlin property (Definition 14.1 in [Phi]), outerness and preservation of simplicity (Lemma 14.7 in [Phi]). Example 3.5 in [Gar1]: Z_2 acts on M_{3^{\infty}} with the tracial Rokhlin property, but not with the Rokhlin property. Crossed products: traces are in bijection with invariant traces (Theorem 13.11 in [Phi]). The crossed product of a TAF algebra by an action with the tracial Rokhlin property is again TAF (including defining TAF); this is Theorem 14.17 in [Phi], and some intermediate results may be taken without proof (for instance Theorem 14.23 in [Phi]). Speakers: Shirley Geffen and Hung Chang Liao.

  • Lectures 13 and 14 (January 25 and February 1): Study the application of the tracial Rokhlin property to crossed products of the canonical finite group actions on the irrational rotation algebra; see [ELPW]. The first talk can focus on showing that the relevant action have the tracial Rokhlin property. The proof in [ELPW] uses several results from von Neumann algebras. Speakers: Eusebio Gardella and N.N.
For help and assistance when preparing the lectures, you may approach Siegfried Echterhoff for lectures 1 through 6, and Eusebio Gardella for lectures 7 through 14.


  • [AS] R. Archbold and J. Spielberg. Topologically free actions and ideals in discrete C*-dynamical systems. Proc. Edinburgh Math. Soc. 37. (1993), 119--124. Available here.
  • [BO] N.P. Brown and N. Ozawa. C*-algebras and finite-dimensional approximation. Graduate Studies in Mathematics 88, AMS. Available here.
  • [Ech] S. Echterhoff. Crossed products and the Mackey-Rieffel-Green machine. Available here.
  • [EL] S. Echterhoff and M. Laca.: The primitive ideal space of the C*-algebra of the affine semigroup of algebraic integers. Math. Proc. Cambridge Philos. Soc. 154. (2013), no. 1, 119--126. Available here.
  • [ELPW] S. Echterhoff, W. Lück, N. C. Phillips, S. Walters, "The structure of crossed products of irrational rotation algebras by finite subgroups of SL_2(Z)". J. Reine Angew. Math. 639 (2010), 173--221. Available here.
  • [Ell] G. Elliott. Some simple C*-algebras constructed as crossed products with discrete outer automorphism groups. Publ. RIMS Kyoto Univ. 16 (1980), 299--311. Available here.
  • [Gar1] E. Gardella. Rokhlin-type properties for group actions on C*-algebras. Lecture notes from a minicourse. Available here.
  • [Gar2] E. Gardella. Compact group actions with the Rokhlin property. To appear in Transactions of the AMS. Available here.
  • [Izu] M. Izumi. Finite group actions on C*-algebras with the Rohlin property, I. Duke Math. J., 122 (2004), no. 2, 233--280. Available here.
  • [Kis] A. Kishimoto. Outer automorphisms and reduced crossed products of simple C*-algebras. Common. Math. Phys. 81 (1981), 429--435. Available here.
  • [Phi] N. C. Phillips. An introduction to crossed product C*-algebras and minimal dynamics. Lecture notes. Available here.
  • [Sie] A. Sierakowski. The ideal structure of reduced crossed products. Münster J. of Math. 3 (2010), 237--262. Available here.