On the topological centre of \(L^{1}(G/H)^{\ast\ast}\) (Q2907027)

Let \(G\) be a locally compact group and \(H\) be a compact subgroup of \(G\). This paper is devoted to show that the Banach algebra \(L^1(G/H)\) is strongly Arens irregular for a large class of locally compact groups.NEWLINENEWLINENEWLINENEWLINEReviewer's remark: The reader(s) should be warned that the paper is full of inaccuracies and statements which are based on mathematically false results. Among the facts which are insufficiently taken into account; (i) Proposition 3.3 of [\textit{R. A. Kamyabi-Gol} and \textit{N. Tavallaei}, Bull. Iran. Math. Soc. 35, No. 1, 129--146 (2009; Zbl 1184.43003)] is simply false for general homogeneous spaces. It can be checked readily that Proposition 3.3 of [loc. cit.] is true if and only if \(H\) is normal in \(G\). (ii) Thus, invoking (i), the convolution (resp. involution) defined in [loc. cit.] are well-defined if and only if \(H\) is a normal subgroup of \(G\) and hence in this case the convolution (resp. involution) coincide with the standard \(\ast\)-algebra structure of the Banach function space \(L^1(G/H)\) via the group structure of \(G/H\). (iii) Concerning the convolution over the Banach measure space \(M(G/H)\), the convolution given by (1.3) is well-defined if and only if \(H\) is a normal subgroup of \(G\), hence in this case the convolution (resp. involution) coincide with the standard \(\ast\)-algebra structure of the Banach measure space \(M(G/H)\) via the group structure of \(G/H\).