Thinking noncommutatively

Abstract

Suddenly non-commutativity is fashionable. Following Connes [Con1, Con2], differential geometers are falling over themselves to go noncommutative. A veritable army of quantum group theorists are deforming coproducts of group algebras and universal enveloping algebras to manufacture new noncommutative mathematical structures. Among the more conservative mathematical disciplines, C*-algebra theorists have known for some time that, following Gel’fand’s characterisation of every commutative unital C*-algebra as a C(X), where X is a compact Haussdorff topological space, if you want to do noncommutative topology you should study noncommutative C*-algebras and particularly their K-theory. And quantum probabilists have long known, following the fact that a commutative von Neumann algebra is essentially an L∞ space of a probability space, that their discipline can be regarded as the study of noncommutative von Neumann algebras. While we welcome the many new enthusiasts for noncommutative structures, perhaps we have some worthwhile advice to offer them based on experience. In particular the present paper is a modest offering of this kind; much of it concerns how to “change the variables” in “functions” which are elements of no longer commutative algebras.