On Connes' joint distribution trick and a notion of amenability for positive maps.
type: Item
instance of: scholarly article
title: On Connes' joint distribution trick and a notion of amenability for positive maps.
zbMATH Open document ID: 1036.46048
author: Sorin Popa
published in: L'Enseignement Mathématique. 2e Série
publication date: 2001-01-30T00:00:00Z
review text: 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 \(G\) [Ma…
Mathematics Subject Classification ID: 46L37
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 43A07
zbMATH DE Number: 1558720
zbMATH Keywords: von Neumann algebras
zbMATH Keywords: amenable linear positive maps
zbMATH Keywords: Følner and Kesten-type amenability conditions
zbMATH Keywords: subfactors
zbMATH Keywords: standard graph
reviewed by: Florin P. Boca
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1558720