Course Description: Crossed Products of C*-algebras and Banach Algebras, University of Toronto, Spring Semester 2014

A crossed product is the functional analysts’ version of a skew group ring. Thus, if α : G→ Aut(A) is an action of a locally compact group G on a Banach algebra A, then a crossed product Banach algebra encodes the action of G on A, and its representation theory is related to pairs (u, π) consisting of a representation u of G and a representation π of A on the same Banach space such that ugπ(a)u −1 g = π(αg(a)) for all g ∈ G and a ∈ A. There is an extensive theory of crossed product C*-algebras, with many active directions of research. Crossed product constructions are a very important way of producing new C*-algebras from old ones. By contrast, crossed products of other sorts of Banach algebras have received very little attention. Very recent work suggests that there are a number of interesting examples. There may be an interesting general theory of crossed products of algebras of operators on L spaces and possibly of other classes of Banach algebra crossed products. This course will primarily be about the structure of C*-algebra and L operator algebra crossed products by discrete groups (although I hope to inspire interest in other classes of Banach algebra crossed products), and to some extent about the classification of simple nuclear C*-algebra arising as crossed products by Z, Z, and finite groups. I will give the basic construction and a number of examples, doing both the C* and L theories in parallel as far as the L theory currently goes. I will continue with further results in the C* theory. I will use the C* results as a source of ideas for possible new research directions for L operator crossed products and to some extent for other kinds of Banach algebra crossed products. The C*-algebra part of the course will follow parts of my (still very incomplete) lecture notes [4]. (I hope to make further progress on them before the course starts, and I expect to continue to update them as the course proceeds.) I know of just one book on crossed products,