Probability measures on groups and some related ideals in group algebras
Let G be a locally compact group, \(\mu\) a probability measure on G, and denote by \(J_{\mu}\) the left ideal in \(L^ 1(G)\) given by the closure of \(\{\) f-f*\(\mu|\) \(f\in L^ 1(G)\}\). This paper studies amenability and bounded \(\mu\)-harmonic functions on G in terms of the structure of these ideals. The author proves that every ideal \(J_{\mu}\) is contained in a maximal one, and G is amenable if and only if there exists a unique maximal ideal of this type (namely, the ideal of \(L^ 1\)-functions with integral equal to zero). Moreover, \(L^ 1(G)/J_{\mu}\) is the predual of the space \(H^{\infty}\) of bounded \(\mu\)-harmonic functions, and there exists a measurable G-space (\(\Omega\),\(\eta\)) such that \(L^{\infty}(\Omega,\eta)\) is isometrically isomorphic with \(H^{\infty}\). If \(J_{\mu}\) is a modular ideal, then the isomorphism is given by the Poisson representation. If \(J_{\mu}\) is order modular, then \(L^ 1(G)\) is the sum of the left and the right principal ideals generated by \(\delta_ e-\mu.\) Although this does not yield an isomorphism between (\(\Omega\),\(\eta\)) and the Poisson boundary of (G,\(\mu\)), it implies that, when \(\mu\) is spread out, there is a continuous surjection from (\(\Omega\),\(\eta\)) to the Poisson boundary, which carries \(\eta\) to the Poisson measure. Maximality of \(J_{\mu}\) is studied in significant examples, which include the free group and \(SL_ 2({\mathbb{R}})\).
scientific article; zbMATH DE number 4198760
scientific article