Event № 119
We will discuss prerequisites for the Operator Algebras Learning Seminar.
C*-algebras constructed out of C*-correspondences have been a central theme in operator algebras for almost twenty years at least. This semester, the seminar will be dedicated to (co)actions on C*-correspondences, (co)actions on the associated algebras and the relations between them.
We will begin with the papers
"Crossed products of C*-correspondences by amenable group actions" by Hao and Ng (JMAA, 2008, link: http://www.sciencedirect.com/science/article/pii/S0022247X08004563 )
followed perhaps by
"Coactions on Cuntz-Pimsner algebras", by Kaliszewski & Quigg & Robertson, (Math. Scand., to appear, link http://arxiv.org/abs/1204.5822)
As these constructions rely on many operator-algebraic notions, we will require a few preliminaries. Most of them will be given during the talks, but in a succinct way. We therefore list several topics, with references, that the audience will be expected to be familiar with - at least at the level of knowing what they mean. We emphasize that up to some preparations, this seminar will be accessible to non-operator algebraists.
* Spatial tensor products of C*-algebras: definition and basics. See the book "Hilbert C*-modules" by Lance, pp. 31-32, or most books on C*-algebras.
* Multiplier algebras: definition and basic theorems. Browse through Chapter 2 of Lance.* Hilbert modules and C*-correspondences: again, definition and basic examples. See "Tensor algebras over C*-correspondences: representations, dilations, and C*-envelopes" by Muhly and Solel (JFA, 1998), Definition 2.1 and the following examples. A more complete overview on Hilbert modules is Lance, Chapter 1.
* Crossed product C*-algebras: we will define them from scratch, but to avoid a shock, it's better to be familiar with this construction. See Chapter 2 of "Crossed Products of C*-Algebras" by D. Williams.
* Cuntz-Pimsner algebras: ditto; look at the definition and see some examples. See Definition 3.5 in "On C*-algebras associated with C*-correspondence" by Katsura (JFA, 2004). For examples, see Muhly-Solel.