Runde and Spronk showed in 2004 that there are non-amenable groups $G$, including $\mathbb F_2$, where $B(G)$ is operator Connes-amenable. This result was surprising since the measure algebra $M(G)$ is Connes-amenable if and only if $G$ is amenable, which might lead one to guess that $B(G)$ should be operator Connes-amenable if and only if $G$ is amenable. This leads to the question: for which groups $G$ is $B(G)$ operator Connes-amenable? We make progress on this problem by giving the first examples of groups where $B(G)$ is known to not be operator Connes-amenable. More specifically, we show that $B(G)$ is not operator Connes-amenable when $G$ is a non-compact locally compact group with property (T) and finite almost periodic compactification, or when $G$ is a discrete group without the factorization property. This refutes the suggestion of Dales that $B(G)$ might be operator Connes-amenable for every locally compact group.
Categories: math.FA math.GR math.OA
