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
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
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
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
After other selected topics, the talks will end by discussing an enticing
list of open problems.
For a highly condensed and entertaining brush-up on functional analysis,
see the pertinent sections of
Nik Weaver, Set theory and C* algebras, The Bulletin of Symbolic Logic 13
(an excellent survey article)
Ilijas Farah, C*-algebras and their representations
(potentially useful since it was written by a set-theorist
struggling to comprehend the key concepts)
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
Gert K. Pedersen, Analysis now, Graduate Texts in Mathematics 118
Springer-Verlag, New York (1989) [LCCN: QA320 P39 1989]
Bruce Blackadar, Operator algebras, Encyclopaedia of Mathematical
Springer-Verlag, Berlin (2006) [ISBN: 978-3540284864]
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.
- The 28X Airport Flyer bus costs $2.60 (no change provided) and departs
every 20-30 minutes. The trip takes about an hour. CMU is the last stop.
- Taxis between the airport and the CMU area cost up to $60. The trip takes about 40 minutes.
- Express Shuttle USA goes between the airport and several local hotels.
Departures every half hour from the airport; buy your ticket at the desk.
Reservations are required from the hotel to the airport; call 412-321-4990.
The trip takes about an hour. The price is $28 one-way and $49 roundtrip.
- Some of the local hotels have shuttles that pick up and
drop off guest at CMU.
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.
- Your name, university affiliation,
mailing address, phone number and email address
- Your professional status and
- undergraduate students: please describe your background in set theory
- graduate students: please tell us your year and the name of
your thesis advisor if you have one
- faculty: please tell us whether you hold a federal research grant
- A brief statement about your interest in the workshop