ABSTRACT
This paper is an expanded version of the lectures I delivered at the Indian Statistical Institute, Bangalore, during the OTOA-2014 conference. My intention at the conference was to offer a gentle introduction to non-selfadjoint operator algebras, using topics that relate to my current research interests. One of my main goals was to provide to the audience most of the prerequisites for understanding the proof of Theorem 2.5.13, which identifies the C * - envelope of a tensor algebra as the corresponding Cuntz-Pimsner C *-algebra28 (providing these prerequisites meant of course that I had to survey a good deal of my operator algebra toolkit). Another goal was to demonstrate (through their classification) that certain non-selfadjoint algebras store a great deal of information about dynamical systems, in a much better way than their C * - counterparts do. These remain two of the main goals of these notes as well. During the preparation of this manuscript however, it occurred to me that there is something else that should be included here. Recently Chris Ramsey and I were able to extend the concept of a crossed product from C *-algebras to arbitrary operator algebras in such a way that many of the selfadjoint results are being preserved in this extra generality, e.g., Takai duality. This opens a new, exciting, and very promising area of research that somehow never attracted the attention it deserved, especially given the applications on C *-algebra theory. (See the discussion after Theorem 2.7.19 in Section 2.7.) Describing these developments has become yet another goal of these notes.
