Abstract Plancherel (Trace) Formulas over Homogeneous Spaces of Compact Groups

This paper introduces a unified operator theory approach to the abstract Plancherel (trace) formulas over homogeneous spaces of compact groups. Let G be a compact group and let H be a closed subgroup of G . Let G/H be the left coset space of H in G and let μ be the normalized G -invariant measure on G/H associated with Weil’s formula. Then we present a generalized abstract notion of Plancherel (trace) formula for the Hilbert space L2(G/H,μ) .