### Topics in Operator Theory, Lecture 5 and on

Last week (which was the fourth week, not really the fourth lecture) we finished the proof of Pick’s interpolation theorem, and then I gave a one hour crash course in C*-algebras. The main topics we covered were:

1. Positive functionals and states on C*-algebras and the GNS construction.
2. For a linear functional $f$ on a C*-algebra, $f\geq 0 \Leftrightarrow f(1) = \|f\|$.
3. The Gelfand-Naimark theorem .
4. A Hahn-Banach extension theorem: If $A$ is a unital C*-algebra and $B$ is a unital C*-subalgebra, then every state on $B$ extends to a state on $A$.

From now on we will begin a systematic study of operator spaces, operator systems, and completely positive maps. I will be following my old notes, which for this part are based on Chapters 2 and 3 from Vern Paulsen’s book , and I will make no attempt at writing better notes.

As I start with some basic things this week, the students should brush up on tensor products of vector spaces and of Hilbert spaces.

UPDATE DECEMBER 4th:

I decided to record here in some more details the material that I covered following Paulsen’s book, since my presentation was not 1-1 according to the book. In what follows, $M$ will denote a unital operator space, $S$ an operator system, and $A$ and $B$ are C*-algebras. Elements in these spaces will be denoted as $a,b$ etc.