Boundary maps for C-crossed products with R with an application to the quantum Hall effect

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with R is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t. Connes’ pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schrödinger operators. 1 Motivation and main result In a commonly used approach to study aperiodic solids, particles in the bulk of the medium are described by covariant families of one-particle Schrödinger operators {Hω}ω∈Ω where Ω is the probability space of configurations furnished with an ergodic action of space translations. Crossed product algebras provide a natural framework for such families [Be86]. In particular their bounded functions are represented by elements of a C-crossed product, the so-called bulk algebra. The non-commutative topology of the C-algebra is a useful tool to construct topological invariants resulting from pairings between K-group elements and higher traces. Some of these invariants may be physically interpreted as topologically quantised quantities; the quantised Hall conductivity is such an example. The physics near a boundary of the solid can also be described by a C-algebra, the so-called edge algebra. The bulk algebra being essentially a crossed product of the edge algebra with R (or with Z in the tight binding approximation [KRS02]), both algebras are tied together in the Wiener-Hopf extension (or respectively the Toeplitz extension). This extension gives rise to boundary maps between the K-groups and the higher traces of the bulk and edge algebra which allow one to equate