Let Hnbe the (2n+1)-dimensional Heisenberg group and Ka compact group of automorphisms of Hnsuch that (K⋉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 Hnand the space of Schwartz function on a closed subset of Rshomeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on Hn.
Astengo, F., DI BLASIO, B., Ricci, F. (2009). Gelfand pairs on the Heisenberg group and Schwartz functions. JOURNAL OF FUNCTIONAL ANALYSIS, 256(5), 1565-1587 [10.1016/j.jfa.2008.10.008].
Gelfand pairs on the Heisenberg group and Schwartz functions
DI BLASIO, BIANCA;
2009
Abstract
Let Hnbe the (2n+1)-dimensional Heisenberg group and Ka compact group of automorphisms of Hnsuch that (K⋉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 Hnand the space of Schwartz function on a closed subset of Rshomeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on Hn.
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