Mathematics – Functional Analysis
Scientific paper
2001-11-19
Math. Proc. Cambridge Phil. Soc. 136 (2004), 675-686
Mathematics
Functional Analysis
15 pages; more typos fixed
Scientific paper
10.1017/S030500410300745X
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.
Runde Volker
Spronk Nico
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