On Connes' joint distribution trick and a notion of amenability for positive maps. (Q1594926)

Let \(P_1\) and \(P_2\) be semifinite von Neumann algebras with normal faithful traces, both of which are denoted by Tr. Let \(\Phi_0,\dots,\Phi_n :P_1 \rightarrow P_2\) be unital, positive, linear maps such that \(\text{Tr} \circ \Phi_j\leq \text{Tr}\) and \[ \sup\{ \| \Phi_j(x)\| _{2,\text{Tr}} \;| \;x\in P_1, \| x\| _{2,\text{Tr}}\leq 1 \},\qquad j=0,1,\dots,n-1. \] Assuming that \(0<\delta <(5n)^{-32}\) and letting \(b_0,b_1,\dots,b_n\) be positive elements in \(P_1\) such that \(\| b_j\| _{2,\text{Tr}}=1\), \(\| \Phi_j (b_j)\| _{2,\text{Tr}} \geq 1-\delta\) and \[ \| \Phi_0(b_0)-\Phi_j(b_j)\| _{2,\text{Tr}} <\delta , \qquad j=0,1,\dots,n-1, \] it is proved that there exists \(s>0\) such that \[ \| \Phi_0 (e_s (b_0))-\Phi_j (e_s(b_j))\| _{2,\text{Tr}} <\delta^{1/4} \| e_s(b_0)\| _{2,\text{Tr}} , \qquad j=1,\dots,n-1, \] where \(e_s (b)\) denotes the spectral projection of \(b\) on the interval \((-\infty,s)\). If, in addition, \(\Phi_0\) is trace preserving, then there exists \(s>0\) such that one also has \[ | \text{Tr} (e_s(b_0))-\text{Tr}(e_s (b_j))| < \delta^{1/6} Tr (e_s(b_0)) \] and \[ \| \Phi_j (e_s (b_j))\| _{2,\text{Tr}} >(1-\delta^{1/32})\| e_s (b_j)\| _{2,\text{Tr}} . \] Taking \(P_1=P_2=P\) and \(\Phi_j\) the identity map on \(P\), one recovers one of the key arguments used by \textit{A. Connes} [Ann. Math. (2) 104, 73--115 (1976; Zbl 0343.46042)] in his remarkable proof of the fact that injective \(II_1\) factors are hyperfinite. Taking \(P_1=P_2 =\ell^\infty (G)\) for \(G\) a discrete group and the counting measure on \(G\) as a trace, and taking \(n=2\) and \(\Phi_0(f)=\Phi_1(f)=\frac{1}{k} \sum_1^k L_{g_i}(f)\) to define the Markov operator with respect to a finite set \(S=\{ g_1,\dots,g_k \}\) of \(G\) which contains the neutral element and \(S=S^{-1}\), one gets an abstract proof for the fact that the spectral condition of \textit{H. Kesten} that the Markov operator has norm one implies the Følner type condition for the amenability of \(G\) [Math. Scand. 7, 146--156 (1959; Zbl 0092.26704)]. In the second part of the paper, the author applies his powerful result to study the amenability of weighted bipartite graphs \(\Gamma\) associated with extremal inclusions of subfactors \(N\subset M\) with Jones index \([M:N]<\infty\). In this setting, he proves that for a large class of such inclusions, the Kesten-type condition \(\| \Gamma \| ^2=[M:N]\) implies a Følner type condition for the graph.