MEDIA RELEASE — November 3, 2010

Canadian Mathematical Society

November 3, 2010

2010 CMS G. de B. Robinson Award Winners

Andrew Toms

Wilhelm Winter

OTTAWA, Ontario — The CMS is pleased to announce that Andrew Toms and Wilhelm Winter are the recipients of the 2010 G. de B. Robinson Award for their paper “$\mathcal{Z}$-Stable ASH Algebras”, published in the Canadian Journal of Mathematics (60:3 2008, 703-720; doi:10.4153/CJM-2008-031-6).

The G. de B. Robinson Award was inaugurated to recognize the publication of excellent papers in the Canadian Journal of Mathematics and the Canadian Mathematical Bulletin and to encourage the submission of the highest quality papers to these journals. The first award was presented in 1996.

"One of the main results of the Toms-Winter paper is to show that every class of $C^*$-algebras which has been classified in this program is $\mathcal{Z}$-stable," stated Ken Davidson, Chair of the CMS Publications Committee. "Until the work of Toms and Winter, almost none of the algebras classified were known to be $\mathcal{Z}$-stable. More generally, they show that every separable, approximately divisible $C^*$-algebra is $\mathcal{Z}$-stable."

The paper makes an important contribution to the (Elliott) program to classify simple, separable nuclear $C^*$-algebras by their K-theoretic invariants. The Jiang-Su algebra (the $\mathcal{Z}$ in the title) is a simple $C^*$-algebra which is K-theoretically equivalent to $\mathbb{C}$. A $C^*$-algebra is $\mathcal{Z}$-stable if it is isomorphic to its tensor product with $\mathcal{Z}$. It is conjectured that the $\mathcal{Z}$-stable, separable, nuclear $C^*$-algebras is the set of $C^*$-algebras which are classifiable by their K-theory.

ASH algebras are $C^*$-algebras which can be obtained as inductive limits of subalgebras of homogeneous $C^*$-algebras, which are spaces of functions with values in a matrix algebra. This technical property has been verified for a wide class of $C^*$-algebras, and it is a key device for many of the deep results in the discipline. In this paper, a large class of ASH algebras including those which were known to be classifiable are shown to be $\mathcal{Z}$-stable, even when they are not approximately divisible.

These results led to more recent work by the authors showing that if $\alpha$ is a minimal homeomorphism of an infinite compact, finite dimensional metric space $X$, then the crossed product $C(X) \times_\alpha \mathbb{Z}$ is $\mathcal{Z}$-stable. This allows the use of K-theory invariants to analyze these dynamical systems.

Andrew Toms was born in Montreal in 1975, and was raised on Prince Edward Island. He attended Queen's University as an undergraduate and obtained his Ph.D. from the University of Toronto in 2002. After holding faculty positions at the University of New Brunswick and York University, he was appointed Associate Professor in the Department of Mathematics at Purdue University in 2010. Toms' mathematical interests include the classification of $C^*$-algebras and points of contact between operator algebras, logic, and topology.

Wilhelm Winter was born in Germany in 1968; he studied mathematics and physics at the Universities of Heidelberg and Muenster, where he received his PhD in 2000. He continued to work in Muenster until his Habilitation in 2006. Since 2007 he holds a lectureship at the University of Nottingham, UK.

The collaboration between Toms and Winter commenced in 2003, and has since resulted in a series of seven joint papers.

For more information, contact:

Prof. Ken Davidson
Chair, CMS Publications Committee
Department of Pure Mathematics
University of Waterloo

About the Canadian Mathematical Society

The Canadian Mathematical Society (CMS) is the main national organization whose goal is to promote and advance the discovery, learning, and application of mathematics. The Society's activities cover the whole spectrum of mathematics including: scientific meetings, research publications, and the promotion of excellence in mathematics education at all levels. The CMS has a number of awards and prizes that are annually given to individuals in recognition of outstanding contributions to the advancement of mathematics.