### 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:

- Positive functionals and states on C*-algebras and the GNS construction.
- For a linear functional on a C*-algebra, .
- The Gelfand-Naimark theorem .
- A Hahn-Banach extension theorem: If is a unital C*-algebra and is a unital C*-subalgebra, then every state on extends to a state on .

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, will denote a unital operator space, an operator system, and and are C*-algebras. Elements in these spaces will be denoted as etc.

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