The Poisson boundary of amenable extensions
Let \(H\) be a closed normal subgroup of a locally compact group \(G\). The author obtains two characterizations of amenability of \(H\) as follows. If \(\pi:G\to G/H\) is the canonical quotient mapping, then the following statements are equivalent: (i) The group \(H\) is amenable; (ii) there exists a Borel probability measure \(\mu\) on \(G\) such that \(\lim_{n\to \infty} \|h\mu^n- \mu^n\|=0\) for all \(h\in H\), where \(\mu^n\) denotes the \(n\)-fold convolution of \(\mu\); (iii) for any Borel probability measure \(\mu'\) on \(G/H\), there exists a Borel probability measure \(\mu\) on \(G\) satisfying the condition in (ii) such that \(\pi(\mu)= \mu'\). Consequently, we can embed the space \(H^\infty (G/H,\mu')\) of bounded \(\mu'\)-harmonic functions on \(G/H\) into the space \(H^\infty (G,\mu)\) of bounded \(\mu\)-harmonic functions on \(G\), which gives a canonical projection of the Poisson boundary \(\Gamma(G,\mu)\) onto the Poisson boundary \(\Gamma(G/H, \mu')\). The second characterization of the amenability of \(H\) is the following: The following statements are equivalent: (i) The group \(H\) is amenable; (ii) there exists a Borel probability measure \(\mu\) on \(G\) such that the Poisson boundaries \(\Gamma(G,\mu)\) and \(\Gamma (G/H, \mu')\) are isomorphic, where \(\mu'= \pi(\mu)\); (iii) for any Borel probability measure \(\mu'\) on \(G/H\), there exists a Borel probability measure \(\mu\) on \(G\) with \(\pi(\mu)=\mu'\) such that the Poisson boundaries \(\Gamma(G,\mu)\) and \(\Gamma (G/H,\mu')\) are isomorphic.
scientific article; zbMATH DE number 1798916