An inner amenable group whose von Neumann algebra does not have property Gamma. (English) Zbl 1250.46041

A \(\mathrm{II}_1\) factor \(M\) with trace \(\tau\) is said to have property \(\Gamma\) if there is a sequence \(( u_n )_{n=1}^\infty\) of unitaries in \(M\) such that \(\tau(u_n) = 0\) for all \(n\) and \(u_n x - xu_n \to 0\) in \(L^2(M,\tau)\) for all \(x \in M\).
A (countable) discrete group \(G\) is called inner amenable if there is a finitely additive set function \(m\) from the subsets of \(G \setminus \{ e \}\) with \(m(G \setminus \{ e \}) = 1\) such that \(m(xAx^{-1}) = m(A)\) for all \(A \subset G \setminus \{ e \}\) and all \(x \in G\): this notion was introduced by E.G.Effros in [“Property \(\Gamma\) and inner amenability”, Proc. Am. Math. Soc. 47, 483–486 (1975; Zbl 0321.22011)].
Let \(LG\) denote the group von Neumann algebra of \(G\), i.e., the von Neumann algebra on \(\ell^2(G)\) generated by all left translations. In [op. cit.], Effros proved that if \(LG\) is a \(\mathrm{II}_1\) factor with property \(\Gamma\), then \(G\) is inner amenable. He also asked if the converse holds true: Does \(LG\) always have property \(\Gamma\) whenever \(G\) is an inner amenable group with infinite conjugacy classes?
After this open problem attracted the attention of many researcher over the years, the author solves it in the present paper.


46L36 Classification of factors
20E45 Conjugacy classes for groups
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L10 General theory of von Neumann algebras
43A07 Means on groups, semigroups, etc.; amenable groups


Zbl 0321.22011
Full Text: DOI arXiv


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