Crossed products of C*-algebras by finite groups (Shanghai, July 2016)

This is the home page for N. C. Phillips' short course "Crossed products of C*-algebras by finite groups" at East China Normal University, Shanghai, 11--29 July 2016.

This course will be on crossed products of C*-algebras by actions of finite groups, with emphasis on the structure of the crossed product when the original algebra is simple and when the action satisfies sufficiently strong freeness conditions (such as the Rokhlin property or the tracial Rokhlin property). Some elementary knowledge of C*-algebras will be assumed, but group actions and crossed products will be defined and treated from scratch. The basics of crossed products by finite groups are particularly elementary, and provide an easy introduction to more general crossed products. To show how the theory fits together, I will sometimes mention what happens in more general situations, but without giving details.

Here are pdf files of the lectures in Shanghai (so far not complete). Some misprint have been corrected; please let me know about any others that you find.

• Lecture 1 (corrected; printable); lecture 1 slides (corrected). Some misprints have been corrected, and the discussion of the last example has been expanded. Second correction: Two exercises added near the end. Third correction: Contents slide corrected.
• Lecture 2 (corrected; printable); lecture 2 slides (corrected). Some misprints corrected, and some additional exercises. Some slides deleted; they will be in lecture 3. The last five slides ("Appendix") were not shown, and won't be in lecture 3. Second correction: Contents slide corrected.
• Lecture 3 (corrected; printable); lecture 3 slides (corrected). Some misprints corrected, and one additional exercise. The slides on C* (G, C (G)), on equivariant homomorphisms and induced maps on crossed products, and on the computation of the crossed product by a specific product type action, have been expanded and split in two slides each. Second correction: A slide added to the end of the appendix; other small changes.
• Lecture 4 (corrected; printable); lecture 4 slides (corrected). Some misprints corrected and a little new material added. (One slide has been split in two.) Second correction: Contents slide corrected. "Standalone versions" (with material on AF algebras omitted from the slides as presented because it is expected to be in other people's lectures): printable; slides.
• Lecture 5 (corrected; printable); lecture 5 slides (corrected). Appendix added with discussion of the situation on type II_1 factors and its relation with the C* case. (From a question asked at the lecture; not shown in any lecture.) Also, some misprints corrected, hint for an exercise added, and other minor changes.
• Lecture 6 (corrected; printable); lecture 6 slides (corrected). Misprints corrected, and other minor changes made. Includes several slides at the end not shown in the lecture.

Here are pdf files of a printable version of the slides for a series of three lectures on a very similar topic at the Research Institute for Mathematical Sciences, Kyoto University, in December 2011:

• Lecture 1.
• Lecture 2.
• Lecture 3.

My draft notes on crossed product C*-algebras and minimal dynamics have a general discussion of crossed products by discrete groups, and several sections devoted to crossed products by actions of finite groups with the Rokhlin property or the tracial Rokhlin property. These notes are still very incomplete.

This page maintained by N. Christopher Phillips, email. Please email plain text (7 bit ASCII) only (no web page coded files, Microsoft Word documents, binary characters, etc.; see the email link for more).

Last significant change 12 July 2016.