Amenability of groups in terms of a perturbation condition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T03:16:51Z http://mathoverflow.net/feeds/question/51680 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51680/amenability-of-groups-in-terms-of-a-perturbation-condition Amenability of groups in terms of a perturbation condition Andreas Thom 2011-01-10T18:51:03Z 2011-02-11T06:20:45Z <p>Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$</p> <p>$$\inf \lbrace\|T-S\| \mid S \in \lambda(G)' \rbrace \leq C \cdot \sup\lbrace \|\lambda(g)T- T \lambda(g) \| \mid g \in G\rbrace.$$</p> <p>(Here $\lambda(G)'$ denotes the commutant of $G$ in $B(\ell^2 G)$.)</p> <blockquote> <p><strong>Question:</strong> Is $G$ amenable?</p> </blockquote> <p>It is fairly easy to see that amenability of $G$ implies the existence of such a constant. Indeed, one may take $S$ to be some fixed point for the conjugation action on $\overline{\rm conv}\lbrace \lambda(g)T\lambda(g)^* \mid g \in G\rbrace$. I am asking for the converse of this statement.</p> <p>EDIT: Since the derivation problem came up in Kate's comment, I want to clarify to what version of it my question is related. The inequality above holds for some $C$ if and only if the first bounded cohomology of $G$ with coefficients in $B(\ell^2 G)$ (with the conjugation action induced by $\lambda$) is reduced. This is a straightforward application of the open mapping theorem. Now, two things are unclear: </p> <blockquote> <p><strong>Question:</strong> Can $H^1_b(G,B(\ell^2 G))$ be reduced without being zero?</p> </blockquote> <p>and</p> <blockquote> <p><strong>Question:</strong> Can $H^1_b(G,B(\ell^2 G))$ be zero without $G$ being amenable?</p> </blockquote> http://mathoverflow.net/questions/51680/amenability-of-groups-in-terms-of-a-perturbation-condition/51686#51686 Answer by Kate Juschenko for Amenability of groups in terms of a perturbation condition Kate Juschenko 2011-01-10T19:43:06Z 2011-01-12T10:17:30Z <p>Let $M=C^*_\lambda(G)''$ be group von Neumann algebra of $G$. The the condition above implies:</p> <p>$d(T, M')\leq C ||ad(T)|_{M}||$ for every $T\in B(l^2 G)$. The last inequality is equivalent to saying that every derivation of $M$ into $B(l^2 G)$ is inner.</p> <p><strong>Edit:</strong> the above inequality is satisfied automatically (was clarified to me by Stuart White).</p> <p>It is known that if $M\subset B(H)$ has a cyclic vector, then every bounded derivation from $M$ into $B(H)$ is inner [E. Christensen, Extensions of derivations II, Math. Scand., 1982]. Thus [Christensen, op cit, Cor 5.4] we have</p> <p>$d(T,M')\leq 3/2\|(\mathrm{ad}| T)_{M}\|$</p> <p>for every $T\in B(l^2 G)$. </p>