The notion of operator amenability was introduced by Z.-J. Ruan in 1995. He
showed that a locally compact group G is amenable if and only if its Fourier
algebra A(G) is operator amenable. In this paper, we investigate 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(F_2) is operator Connes-amenable
although F_2, the free group in two generators, fails to be amenable.Comment: 15 pages; more typos fixe
