On Isomorphism Between Certain Group Algebras on the Heisenberg Group

Let $I\!\!H_n$ denote the $(2n+1)$-dimensional Heisenberg group and let $K$ be a compact subgroup of $Aut(I\!\!H_n)$, the group of automorphisms of $I\!\!H_n$. We prove that the algebra of radial functions on $I\!\!H_n$ and the algebra of spherical functions arising from the Gelfand pairs of the form $(K, I\!\!H_n)$ are algebraically isomorphic.