Gelfand pairs on the Heisenberg group and Schwartz functions

Details Gelfand pairs on the Heisenberg group and Schwartz functions Gelfand pairs on the Heisenberg group and Schwartz functions

2008-05-25

arxiv.org/abs/0805.3809v1

Mathematics

Functional Analysis

28 pages

Scientific paper

Let $\Hn$ be the $(2n+1)$-dimensional Heisenberg group and $K$ a compact group of automorphisms of $\Hn$ such that $(K\ltimes \Hn,K)$ is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of $K$-invariant Schwartz functions on $\Hn$ and the space of Schwartz function on a closed subset of $\R^s$ homeomorphic to the Gelfand spectrum of the Banach algebra of $K$-invariant integrable functions on $\Hn$.

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