Homological properties of Fourier algebras on homogeneous spaces
After the pioneering work of B. E. Johnson, J. L. Taylor and A. Y. Helemskii on the homological properties of the Fourier algebra \(A(G)\), the topic again got revived through the recent papers of Wood, Forrest, Spronk, Samei, Runde and others. Biprojectivity, biflatness and the operator analogues of these have been studied for the Fourier algebra \(A(G)\) of a locally compact group \(G\). In this interesting note the authors carry out the analogous questions to the Fourier algebra \(A(G/K)\), where \(K\) is a compact subgroup of the locally compact group \(G\). The authors prove that \(A(G/K)\) is operator biprojective if and only if \(K\) is open (or equivalently \(G/K\) is discrete), in Section 4. Completely complemented ideals in \(A(G/K)\) are characterised in Section 5. In the cited work \textit{V. Runde} [Arch. Math. 92, 525--530 (2009; Zbl 1178.22009)] has given a necessary condition for \(A(G)\) to be biflat and biprojective. The authors extend these results to \(A(G/K)\) in the last section of the paper. In particular the authors prove that if \(A(G/K)\) is operator projective then \(K\) is open and in this case \(G\) does not contain the free group on two generators as a closed subgroup.
scientific article; zbMATH DE number 5902950
scientific article