3 - Set theory and operator algebras

Summary

The fifth Appalachian Set Theory workshop was held at Carnegie Mellon University on February 9, 2008. The lecturer was Ilijas Farah. As an undergraduate student Eric Wofsey assisted in writing this chapter, which is based on the workshop lectures.

These notes are based on the six-hour Appalachian Set Theory workshop given by Ilijas Farah on February 9th, 2008 at Carnegie Mellon University. The first half of the workshop (Sections 1–4) consisted of a review of Hilbert space theory and an introduction to C*-algebras, and the second half (Sections 5–6) outlined a few set-theoretic problems relating to C*-algebras. The number and variety of topics covered in the workshop was unfortunately limited by the available time.

Good general references on Hilbert spaces and C*-algebras include [8], [13], [17], [51], [61], and [67]. An introduction to spectral theory is given in [9]. Most of the omitted proofs can be found in most of these references. For a survey of applications of set theory to operator algebras, see [64].

Acknowledgments

We owe greatest thanks to Ernest Schimmerling, without whose enthusiasm and support this paper would not have existed. We would like to thank NikWeaver for giving us a kind permission to include some of his unpublished results. I.F. would like to thank George Elliott, N. Christopher Phillips, Efren Ruiz, Juris Steprāns, Nik Weaver and the second author for many conversations related to the topics presented here.