Operator amenability of Fourier–Stieltjes algebras

Abstract

We investigate, for a locally compact group $G$, the operator amenability of the Fourier–Stieltjes algebra $B(G)$ and of the reduced Fourier–Stieltjes algebra $B_r(G)$. The natural conjecture is that any of these algebras is operator amenable if and only if $G$ is compact. We partially prove this conjecture with mere operator amenability replaced by operator $C$-amenability for some constant $C\,{<}\, 5$. In the process, we obtain a new decomposition of $B(G)$, which can be interpreted as the non-commutative counterpart of the decomposition of $M(G)$ into the discrete and the continuous measures. We further introduce a variant of operator amenability – called operator Connes-amenability – which also takes the dual space structure on $B(G)$ and $B_r(G)$ into account. We show that $B_r(G)$ is operator Connes-amenable if and only if G is amenable. Surprisingly, $B({\mathbb F}_2)$ is operator Connes-amenable although ${\mathbb F}_2$, the free group in two generators, fails to be amenable.