Harmonic Functions on Groups and Fourier Algebras, Issue 1782
Springer Science & Business Media, May 27, 2002 - Mathematics - 100 pages
This research monograph introduces some new aspects to the theory of harmonic functions and related topics. The authors study the analytic algebraic structures of the space of bounded harmonic functions on locally compact groups and its non-commutative analogue, the space of harmonic functionals on Fourier algebras. Both spaces are shown to be the range of a contractive projection on a von Neumann algebra and therefore admit Jordan algebraic structures. This provides a natural setting to apply recent results from non-associative analysis, semigroups and Fourier algebras. Topics discussed include Poisson representations, Poisson spaces, quotients of Fourier algebras and the Murray-von Neumann classification of harmonic functionals.
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action of G adapted algebra amenable AP(G Banach algebra bounded called closed constant contains continuous character contractive projection converges convex hull Ctu(G define denote dense distal dual equicontinuous equivalent Example exists extreme point f x g follows Fourier algebra functions on G G acts transitively G M(G Given gives group action group G harmonic functions Hence homeomorphism idempotent identity implies induces isometry isomorphic Jordan Lemma Let G Ll(G locally compact group means measure on G multiplicative neighbourhood Neumann algebra non-degenerate periodic pointwise Poisson representation Poisson space probability measure Proof properties Proposition Remark restriction result semigroup semigroup structure shown subgroup subnet supp Theorem topology translation invariant uniformly continuous WAP(G weak weak*-topology weakly write x G G