Lebesgue weighted \(L^p\)-algebra on locally compact groups
For a locally compact group \(G\) let \(\omega\) be a positive measurable function on \(G\) of moderate growth, i.e. satisfying \(\text{ess}\sup_y\frac{\omega(xy)}{\omega(x)\omega(y)}<\infty\) for all \(x\in G\). Let \(L^p(G,\omega)\) be the space of measurable functions \(f\) with \(\omega f\in L^p(G)\), norm \(\| f\| _{p,\omega}=\| \omega f\| _p\) and \(\mathcal L_\omega^{1,p} :=L^1(G)\cap L^p(G,\omega)\) which is a Banach space with norm \(\| | f| \| :=\| f\| _1+\| f\| _{p,w}\). The author deals with the existence of (bounded) left approximate identities on spaces \(\mathcal L_\omega^{1,p}\) for which this space is a Banach algebra under convolution and \(\omega\) is of moderate growth and applications to amenability. Some statements are misleading and proofs are hard to follow. In Proposition 2.1 it is claimed that \(\mathcal L_\omega^{1,p}\) is a Banach algebra iff it is an algebra, but the proof only gives that for a fixed function \(\omega\). This is true for a suitably normalized Haar measure with an essential step in the proof given by ``we gave a proof in [the author et al., Publ. Math. 75, No. 3--4, 365--374 (2009; Zbl 1224.43003)] in some cases''. Proposition 2.4 states that for \(1<p<\infty\), \(\mathcal L_\omega^{1,p}\) has a bounded left approximate identity iff \(G\) is discrete. The proof uses a weak*-cluster point of an approximate identity in \(\mathcal L_\omega^{1,p}\). The dual of this space was shown to be \(L^\infty(G)+L^q(G,1/\omega)\) but it is not explained why \(\mathcal L_\omega^{1,p}\) is the dual of some Banach space.
scientific article; zbMATH DE number 6006187
scientific article