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

Post-workshop materials