Fourier analysis on multiplier representations of locally compact Abelian groups

Abstract

Generalising known results [2] for vector groups, it is shown that, for an arbitrary multiplier ω for an arbitrary locally compact Abelian group G, there is a faithful normal semifinite trace on the von Neuman algebra generated by the regular ω-representation of G which is translation invariant in a certain sense. Analogues of the Fourier transformation, the Plancherel identity and the Fourier inversion formula are obtained in which this trace replaces the Haar measure on the dual group.