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A non-probabilistic approach to Poisson spaces

Published online by Cambridge University Press:  14 November 2011

Alan L. T. Paterson
Alan L. T. Paterson
Affiliation:
Department of Mathematics, University of Aberdeen
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Synopsis

Using techniques from probability theory, it has been established that if μ is a probability measure on a separable, locally compact group, then the space of μ-harmonic functions on the group can be identified with C(X) for some compact, Hausdorff space X. The space X is known as the Poisson space of μ. We generalise this result in the context of a measure μ on a locally compact semigroup S, in particular establishing the existence of a Poisson space for non-separable groups. The proof is non-probabilistic, and depends on properties of projections on C(K)(K compact Hausdorff). We then show that if S is compact and the support of μ generates S, then the Poisson space associated with μ, is X, where X×G×Y is the Rees product representing the kernel of S.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 93 , Issue 3-4 , 1983 , pp. 181 - 188
Copyright
Copyright © Royal Society of Edinburgh 1983

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