Operator Algebras

Multi Variable Operator Theory with Relations

Speaker: 
Ken Davidson
Date: 
Tue, May 24, 2011
Location: 
PIMS, University of Victoria
Conference: 
Canadian Operator Symposium 2011 (COSY)
Abstract: 
TBA

Aperiodicity: Lessons from Various Generalizations

Author: 
C. Radin
Date: 
Thu, Aug 1, 2002
Location: 
University of Victoria, Victoria, Canada
Conference: 
Aperiodic Order, Dynamical Systems, Operator Algebras and Topology
Abstract: 
This is a slightly expanded version of a talk given at the Workshop on Aperiodic Order, held in Victoria, B.C. in August, 2002. The general subject of the talk was the densest packings of simple bodies, for instance spheres or polyhedra, in Euclidean or hyperbolic spaces, and describes recent joint work with Lewis Bowen. One of the main points was to report on our solution of the old problem of treating optimally dense packings of bodies in hyperbolic spaces. The other was to describe the general connection between aperiodicity and nonuniqueness in problems of optimal density.

Projective Modules in Classical and Quantum Functional Analysis

Author: 
A. Ya. Helemskii
Date: 
Mon, Aug 11, 2003
Location: 
University of Alberta, Edmonton, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
Homological theory of the “algebras in analysis” exists in at least three different versions. First of all, there is the homological theory of Banach and more general locally convex algebras. This is about 40 years old. However, in the last decade of the previous century, a “homological section” appeared in a new branch of analysis, the so-called quantized functional analysis or, more prosaically, the theory of operator spaces. One of principal features of this theory, as is now widely realized, is the existence of different approaches to the proper quantum version of a bounded bilinear operator. In fact, two such versions are now thought to be most important; each of them has its own relevant tensor product with an appropriate universal property. Accordingly, there are two principal versions of quantized algebras and quantized modules, and this leads to two principal versions of quantized homology. Thus we have now, in the first decade of the 21st century, three species of topological homology: the traditional (or “classical”) one, and two “quantized” ones. In these lectures, we shall restrict ourselves by studying, in the framework of these three theories, the fundamental concept of a projective module. This concept is “primus inter pares” among the three recognized pillars of the science of homology: projectivity, injectivity, and flatness. It is this notion that is the cornerstone for every sufficiently developed homological theory, let it be in algebra, topology, or, as in these notes, in functional analysis. Our initial definitions of projectivity do not go far away from their prototypes in abstract algebra. However, the principal results concern essentially functional-analytic objects. As we shall see, they have, as a rule, no purely algebraic analogues. Moreover, some phenomena are strikingly different from what algebraists could expect, based on their experience.
Notes: 

Actions of Z^k associated to higher rank graphs

Author: 
U. Kumjian,
D. Pask
Date: 
Thu, Aug 1, 2002
Location: 
University of Victoria, Victoria, Canada
Conference: 
Aperiodic Order, Dynamical Systems, Operator Algebras and Topology
Abstract: 

We construct an action of $ \mathbb Z^k $ on a compact zero-dimensional space obtained from a higher graph $ \Lambda $ satisfying a mild assumption generalizing the construction of the Markov shift associated to a nonnegative integer matrix. The stable Ruelle algebra $ R_s(\Lambda) $ is shown to be strongly Morita equivalent to $ C^*(\Lambda) $. Hence $ R_s(\Lambda) $ is simple, stable and purely infinite, if $ \Lambda $ satisfies the aperiodicity condition.

Published in: Ergodic Theory Dynam. Systems 23 (2003), no. 4, 1153-1172.
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