Operator amenability of Fourier–Stieltjes algebras

In this paper, we investigate, for a locally compact groupG, the operator amenability of the Fourier-Stieltjes algebra B(G) and of the reduced Fourier-Stieltjes algebra Br(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 Br(G) into account. We show that Br(G) is operator Connes-amenable if and only if G is amenable. Surprisingly, B(F2) is operator Connes-amenable although F2, the free group in two generators, fails to be amenable.