Gelfand pairs on the Heisenberg group and Schwartz functions

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