The Poisson flow associated with a measure
Let T be a locally compact topological group and let \({\mathcal R}\) be the Banach algebra of bounded right uniformly continuous functions on T. Let \(\mu\) be a regular Borel probability measure on T (\(\mu\in M(T))\). Denote by \(| {\mathcal R}|\) the Gelfand space of \({\mathcal R}\). It can be shown that \(| {\mathcal R}|\) contains T as a dense subset, (\(| {\mathcal R}|,T)\) is a right transformation group and \(| {\mathcal R}|\) has a semi-group structure which induces a semi-group structure on M(\(| {\mathcal R}|)\) (the regular Borel probability measures on \(| {\mathcal R}|)\) via convolution. A function \(f\in {\mathcal R}\) is said to be \(\mu\)-harmonic if \(\int_{| {\mathcal R}|}f(t't)du(t')=f(t)\) (t\(\in T)\). Let \({\mathcal H}_{\mu}\) be the set of \(\mu\)-harmonic functions in \({\mathcal R}\). Let \(\mu\in M(T)\). By Kakutani's fixed point theorem there is an idempotent measure \(\nu\) \((\nu^ 2=\nu)\) on \(| {\mathcal R}|\) such that \(\nu \cdot \mu =\nu\) and \({\mathcal H}_{\nu}={\mathcal H}_{\mu}\). If the support of \(\mu\) is all of T, then the support S of \(\nu\) is a subflow of \(| {\mathcal R}|\) and the restriction R: \({\mathcal R}\to C(S)\) maps the set \({\mathcal H}_{\mu}\) of \(\mu\)-harmonic functions isometrically onto a uniformly closed T- invariant subalgebra \({\mathcal H}_{\mu}\) of C(S). The Poisson boundary of \(\mu\) is the Gelfand space of the image of \({\mathcal H}_{\mu}\) in C(S). It is shown that \(B=\{\nu \cdot \rho | \quad \rho \in S\}\). The representing measure \(\omega\in M(B)\) is any \(\omega\) such that \(b(\omega)=\nu\) where b: M(B)\(\to closed\) convex hull of B, is the barycenter mapping. Let \(t\in T\). Then \(\omega\) is regular at t if there exists \(\delta >0\) such that C closed with \(\omega (C)>1-\delta\) implies that \(C\cap Cs\neq \emptyset\) for all s in some neighborhood \(U_ c\) of t. The authors prove the following theorem. Let \((r_ n)\), \((s_ n)\), \((t_ n)\) be sequences such that \((s_ n)\), \((r_ n)\) converge to \(t_ ns_ n=r_ nt_ n\) for all n, \(\omega\) regular at r and \(xt_ n\to b\) (x\(\in A)\) with \(\omega (A)=a\). Then \(bs=b.\) This result is used to give elegant proofs of several deep results in boundary theory. For example the only \(\mu\)-harmonic functions for a symmetric measure \(\mu\) absolutely continuous with respect to Haar measure on a solvable Lie group are the constants. Another consequence is that using the structure of the Lie algebra of a connected semi-simple Lie group with finite center and no-compact factors it is shown that T acts transitively on the Poisson boundary of any \(\mu\in M(T)\) such that \(\mu\) is absolutely continuous and \(\sup p(\mu)=T\).
scientific article; zbMATH DE number 4074023