2005–2006 Erdős Colloquium:
William Arveson
Operator theory and the K-homology of algebraic varieties
Speaker: William Arveson, University of California, Berkeley
Date and time: Thursday, March 2, 2006, 4:00–5:00 p.m.
Location: Reitz Union, Room 282
Opening remarks: Neil Sullivan, Dean of the College of Liberal Arts and Sciences
Refreshments: 3:30 p.m., before the lecture
Abstract
Let \(X, Y, Z\) be three mutually commuting operators acting on a common Hilbert space that satisfy a nonlinear equation of the form \(X^n + Y^n = Z^n\) for some \(n=2,3,\ldots\). The \(C^*\)-algebra generated by \(X,Y,Z\) is typically noncommutative and can be viewed as a nonclassical counterpart of the curve \(V \subseteq \mathbb{C}^3\) defined by \(x^n+y^n=z^n\). Similarly, there are natural nonclassical counterparts of more general algebraic varieties \(V \subseteq \mathbb{C}^d\).
Starting from first principles, we describe a natural construction of universal operator solutions of equations like this one, and we describe the general properties of these operator solutions, focusing on the question: When does an operator solution of such a system determine an element of the K-homology of the associated classical variety \(V\)? We formulate this question as a concrete conjecture about self-commutators—such as \(X^*X – XX^*\), \(X^*Y – YX^*\), and others in the example above—and describe recent progress on proving the conjecture in general.
About the speaker
William Arveson is professor of mathematics at the University of California, Berkeley. He has held numerous visiting positions and fellowships including Newcastle (UK), Aarhus, Rio de Janeiro, Oslo, UC San Diego, Nankai, Canberra, Penn, Trondheim, Kyoto, two years (1985–86 and 1999–00) as Miller research professor at Berkeley. His theory of extensions of completely positive maps now permeates the study of operator algebras. A current interest is the study of endomorphisms of operator algebras (\(E_0\)-semigroups), which models non-commutative dynamics arising in quantum theory.