Analytic K-theory

The mathematical objects known as C*-algebras dates from fundamental work of the 1940s. A C*-algebra can be described by axioms as a complete complex normed vector space equipped with a multiplication and an involution, where the norm satisfies the C*-identity. One example of a C*-algebra is the algebra of continuous linear maps on a Hilbert space, that is a complete inner product space. The involution arises from taking the adjoint of a map.

Gelfand and Naimark's work gave us two remarkable results on C*-algebras. The first was that any C*-algebra where the multiplication is commutative is isomorphic to the algebra of continuous maps, vanishing at infinity, from some topological space, depending on the algebra, to the complex numbers. The second result is that any C*-algebra is isomorphic to a closed subalgebra of the continuous linear maps on a Hilbert space.

Both results are beautiful, and key to modern research in C*-algebras, and linking them to other areas of mathematics. The second of these results gives us a fundamental connection between work in C*-algebras and in operator theory. In quantum physics, and we will return to this point below, observables are considered operators on a Hilbert space, meaning C*-algberas are objects used to describe quantum systems.

Non-commutative geometry relies on the first of these results; certain invariants of spaces, most notably K-theory, still work as invariants of C*-algebras, and certain geometric structures are better viewed as C*-algebras than as spaces. This point of view includes index theory and the Baum-Connes conjecture, which links topology and representation theory in a fundamental way, and has applications to geometric topology and the structure of C*-algebras.

Recently, Mitchener developed a notion known as LK*-categories. These are logical, if more axiomatically complicated, generalisations of C*-algebras. The fundamental example of an LK*-category is the category of all linear maps (not necessarily continuous) between subspaces of a given Hilbert space. The inspiration here is quantum physics; fundamental observables, such as position and momentum, are maps of this type, and are not continuous. The key result on LK*-categories is that any LK*-category is a category of linear maps between subspaces of a Hilbert space, a connection analogous to that between a C*-algebra and continuous maps on a Hilbert space.

The project focuses on further work in LK*-categories. In principle, they would be a highly suited framework in which to study quantisation, part of mathematics inspired by the passage from classical to quantum mechanics. The main thrust is to develop K-theory for LK*-categories in a way that generalises C*-algebra K-theory, and to develop isomorphism conjectures along the lines of the Baum-Connes conjecture, and various analogues of that conjecture in coarse geometry. In particular, it may be possible to handle coarse conjectures using general machinery.