Appalachian set theory

Appalachian set theory

Saturday, February 9, 2008

9:30 a.m. - 6 p.m. with coffee and lunch breaks

Carnegie Mellon University

Baker Hall A53

Ilijas Farah : "Set theory and operator algebras"


Some long-standing problems from the theory of C* algebras were recently solved by using increasingly sophisticated set theoretic tools. I will present a forward-looking overview of this newly emerging subject.

Only the most basic knowledge of functional analysis will be assumed. In the first lecture I will go over the basics of operators and operator algebras on a (complex) Hilbert space and in particular the spectral theory and GNS representations. Here GNS (Gelfand-Naimark-Segal) stands for an operator-algebraist's way of saying that the class of concrete C* algebras is axiomatizable in an appropriate logic.

A special attention will be given to the Calkin algebra, C(H), on a separable infinite-dimensional Hilbert space H. The Calkin algebra is the quotient of the algebra of all bounded linear operators on H over its ideal of compact operators.

After studying the lattice of projections in C(H) and showing some of its amusing properties (e.g., that it is not a lattice) we shall move to automorphisms. I will construct an outer automorphism of the Calkin algebra using the Continuum Hypothesis (Phillips-Weaver) and outline the fact that assuming Todorcevic's (open coloring) Axiom all automorphisms are inner (Farah).

I will also construct (using less than CH) a pure state on the algebra of bounded operators on a Hilbert space not diagonalizable by any abelian subalgebra (Akemann-Weaver).

After other selected topics, the talks will end by discussing an enticing list of open problems.

Suggested reading

For a highly condensed and entertaining brush-up on functional analysis, see the pertinent sections of Browsing any of the excellent books available on the subject of C* algebras could also be beneficial; an up-to-date, comprehensive and insightful survey is

Lodging, local transportation and parking

The following lodging establishments offer discounted rates for visitors to the Department of Mathematical Sciences at CMU. Here are some options for traveling between the airport, CMU and your hotel. On weekends, parking is free at CMU in the East Campus Garage.

Participant travel support

Funds provided by the National Science Foundation will be used to reimburse some participant transportation and lodging expenses. Priority will be given to students and faculty who do not hold federal research grants. Please request such funds as far in advance of the meeting as possible by sending the following information to James Cummings and Ernest Schimmerling by email.