Intermediate subalgebras and bimodules for general crossed products of von Neumann algebras

Let [Formula: see text] be a discrete group acting on a von Neumann algebra [Formula: see text] by properly outer ∗-automorphisms. In this paper, we study the containment [Formula: see text] of [Formula: see text] inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the [Formula: see text]-bimodules that are closed in the Bures topology and which coincide with the [Formula: see text]-closed ones under a mild hypothesis on [Formula: see text]. We use these results to obtain a general version of Mercer’s theorem concerning the extension of certain isometric [Formula: see text]-continuous maps on [Formula: see text]-bimodules to ∗-automorphisms of the containing von Neumann algebras.

Fredholm theory for arbitrary measure spaces

The classical formulae for Fredholm integral equations, including expansions in terms of eigenfunctions such as Mercer's Theorem are extended to square-integrable kernels on an arbitrary measure space.

Mercer's Theorem and Fredholm resolvents

Multivariate versions of Mercer's Theorem and the usual expansions of the resolvent and Fredholm determinant are shown to hold for an n × n symmetric kernel N(x, y) with arbitrary domain in Rp under weakened continuity conditions. Further, the resolvent and determinant of N(x, y) − a(x)b(y) are given in terms of those of N(x, y).