A short proof of the approximation conjecture for amenable groups
A standing conjecture, viz. Approximation Conjecture, in the theory of \(L^2\)-invariants, for \((R,G)\) where \(G\) is a discrete group has been proven in various cases and versions; especially the linearly ordered version of the conjecture for arbitrary group \(G\) and those directed systems \((G_i)_{i\in \mathbb{N}}\) which consist of finite index subgroups with integer coefficients \(R=\mathbb{Z}\) has been treated by numerous authors. In this paper, the author gives a short proof of Approximation Conjecture with complex coefficients for amenable groups; it is to be noted that a discrete group \(G\) is called amenable if there exists a linear functional \(m:l^\infty (G;R)\to R\) satisfying \(\inf\{\xi(g):g\in G\}\leq m(\xi)\leq \sup\{\xi(g):g\in G\}\).
scientific article; zbMATH DE number 5376860