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.

- Nik Weaver, Set theory and C* algebras, The Bulletin of Symbolic Logic 13 (2007), 1-20 [Link] (an excellent survey article)
- Ilijas Farah, C*-algebras and their representations [PDF] (potentially useful since it was written by a set-theorist struggling to comprehend the key concepts)

- 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 Sciences 122 Springer-Verlag, Berlin (2006) [ISBN: 978-3540284864]

- Slides from the lectures
- Lecture notes by Ilijas Farah and Eric Wofsey

- The
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- faculty: please tell us whether you hold a federal research grant

- A brief statement about your interest in the workshop