This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ a closed subgroup of $G$. Let $\mu $ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated with Weil's formula and $1\,\le \,p\,<\,\infty $. We then present a structured class of abstract linear representations of the Banach convolution function algebras ${{L}^{p}}\left( G/H,\,\mu \right)$.

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 $\mu$ 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 ${{L}^{2}}\left( {G}/{H,\,\mu }\; \right)$.

This paper presents a structured study for abstract harmonic analysis of relative convolutions over canonical homogeneous spaces of semidirect product groups. Let $H,K$ be locally compact groups and $\unicode[STIX]{x1D703}:H\rightarrow \text{Aut}(K)$ be a continuous homomorphism. Let $G_{\unicode[STIX]{x1D703}}=H\ltimes _{\unicode[STIX]{x1D703}}K$ be the semidirect product of $H$ and $K$ with respect to $\unicode[STIX]{x1D703}$ and $G_{\unicode[STIX]{x1D703}}/H$ be the canonical homogeneous space (left coset space) of $G_{\unicode[STIX]{x1D703}}/H$. We present a unified approach to the harmonic analysis of relative convolutions over the canonical homogeneous space $G_{\unicode[STIX]{x1D703}}/H$.