Essential spectrum and fredholm properties for perators on locally compact groups

Abstract

We study the essential spectrum and Fredholm properties of certain integral and pseudo-differential operators associated to non-commutative locally compact groups G. The techniques involve crossed product C*-algebras. We extend previous results on the structure of the essential spectrum to self-adjoint operators belonging (or affiliated) to the Schrodinger representation of certain crossed products. When the group G is unimodular and type I, we cover a new class of global pseudo-differential differential operators with operator-valued symbols involving the unitary dual of G. We use recent results of Nistor, Prudhon and Roch on the role of families of representations in spectral theory and the notion of quasi-regular dynamical system.