From eschimme@andrew.cmu.edu Mon Nov 26 14:48:38 2007 Date: Mon, 26 Nov 2007 14:48:38 -0500 (EST) From: Ernest Schimmerling To: eschimme@math.cmu.edu Subject: abstract and a photo (fwd) ---------- Forwarded message ---------- Date: Sat, 24 Nov 2007 16:15:08 -0500 From: Ilijas Farah To: Ernest Schimmerling Subject: abstract and a photo Hi Ernest, Here is my abstract and here is my photo. I hope that both are acceptable. I did not use TeX, so some of the information in the abstract is to some extent lost. C in C(H) and C in C(X) should be in different fonts; but this is not really a big deal. Best wishes, Ilijas 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 review basics on operators and operator algebras on a (complex) Hilbert space and in particular the spectral theory and GNS representations. A special attention will be given to the Calkin algebra, C(H). It is generally considered to be a quantized analogue of P(N)/Fin. (To an untrained eye this appears as a misnomer, since C(H) is obviously a quantized analogue of C(N*), the C* algebra of continuous complex-valued functions on the Cech-Stone remainder N* of the natural numbers. However, a compact Hausdorff space X is interchangeable with C(X) and via the topological duality P(N)/Fin and N* are essentially the same object.) 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 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 Nik Weaver, Set theory and C* algebras, The Bulletin of Symbolic Logic 13 (2007), 1-20 (an excellent survey article). Ilijas Farah, C*-algebras and their representations, available at http://www.math.yorku.ca/~ifarah/notes.html (potentially useful since it was written by a set-theorist struggling to comprehend the key concepts). For a highly condensed and entertaining brush-up on functional analysis, see the pertinent sections of Pedersen, G.K. Analysis now, Graduate Texts in Mathematics 118 Springer-Verlag, New York (1989) 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 Blackadar, B. Operator algebras, Encyclopaedia of Mathematical Sciences 122 Springer-Verlag, Berlin (2006) [ Part 2, "" Image/JPEG 47KB. ] [ Unable to print this part. ]