Projectivity of modules over Segal algebras (Q2839884)

The Fourier algebra \(A(G)\) of a locally compact group \(G\) is a commutative Banach algebra related to its corepresentation theory. It is isomorphic to the Banach algebra \(L^1(\hat{G})\) of the dual group if \(\hat{G}\) is Abelian.NEWLINENEWLINEIn a previous article [Proc. Lond. Math. Soc. (3) 102, No. 4, 697--730 (2011; Zbl 1229.43005)], the authors studied whether certain naturally defined operator modules over the Fourier algebra are projective or injective. This article continues these investigations, replacing the Fourier algebra by related operator algebras. Here, the term operator algebras means the modules are operator spaces, that is, they come with a system of norms on matrix spaces over them.NEWLINENEWLINEThe article considers, on the one hand, the algebra \(L^1(G)\cap A(G)\) and the Feichtinger algebra; both are dense ideals in \(A(G)\). On the other hand, it considers the closure \(A_{\mathrm{cb}}(G)\) of the Fourier algebra in the algebra of completely bounded multipliers of \(A(G)\).NEWLINENEWLINEThis article contains many negative results: certain naturally defined operator modules like the full and reduced group C*-algebras are never operator projective over \(L^1(G)\cap A(G)\) or the Feichtinger algebra, unless \(G\) is finite. Some of the modules over \(A_{\mathrm{cb}}(G)\) are operator projective if \(G\) is discrete and amenable or if \(G\) is discrete and weakly amenable with residually finite group C*-algebra.