Conditional expectations on \(CV_{p}(G)\). Applications
For a locally compact group \(G\), let \(A_p(G)\) be the Figà-Talamanca-Herz algebra and let \(CV_p(G)\) be the Banach algebra of \(p\)-convolution operators on \(L^p(G)\). For any closed subgroup \(H\) of \(G\), the author finds a condition on the pair of \((G,H)\), denoted by \(G\in [{\mathcal A}]_H\), which implies the existence of a conditional expectation \({\mathcal P}\) of \(CV_p(G)\) onto \(\{T\in CV_p(G):\text{supp}\,T\subset H\}\), i.e., a projection \({\mathcal P}\) from \(CV_p(G)\) onto \(\{T\in CV_p(G): \text{supp}\,T\subset H\}\) such that \({\mathcal P}(uT)=u{\mathcal P} (T)\) for \(u \in A_p(G)\) and \(T\in CV_p(G)\). In addition, this projection satisfies \({\mathcal P} (T)\in PM_p(G)\) if \(T\in PM_p(G)\) and \({\mathcal P} (\mu)=1_H \mu\) if \(\mu\) is a bounded measure. It is proved in this paper that if \(H\) is neutral in \(G\) or \(H\) is amenable, then \(G\in[{\mathcal A}]_H\). As a consequence the author obtains that for \(G\in[{\mathcal A}]_H\), \(H\) is locally \(p\)-Ditkin in \(G\). This generalizes several existing results. Another consequence is that for \(G\in[{\mathcal A}]_H\), the Banach algebra \(cv_p(H)\) is canonically isometrically isomorphic to \(\{T\in cv_p (G): \text{supp}\,T\subset H\}\), where \(cv_p(H)\) \([cv_p(G)]\) is the norm closure of the set of all convolution operators in \(CV_p(H)\) \([CV_p(G)]\) with compact support. This generalizes a theorem of Reiter for abelian locally compact groups \(G\).
scientific article; zbMATH DE number 5167469
scientific article