Harmonic Analysis on the Heisenberg Group pp 109-154 | Cite as

# Group Algebras and Applications

## Abstract

Even though the algebra *L*^{ 1 } (*H* ^{ n }) is not commutative, the subalgebra *L*^{ 1 } (*H*^{ n }/*U(n)*) of radial functions forms a commutative Banach algebra under convolution. In this chapter we study the Gelfand transform on this algebra. The Gelfand spectrum is identified with the set of all bounded *U*(*n*)-spherical functions which are given by Bessel and La-guerre functions. We also consider the Banach algebra generated by the surface measures µ_{r} and get optimal estimates for its characters, from which we proceed to study Wiener-Tauberian theorems and spherical means. We prove a one radius theorem for the spherical means using the summability result of Strichartz proved in the previous chapter. We also prove a maximal theorem for the spherical means on the Heisenberg group.

## Keywords

Banach Algebra Heisenberg Group Group Algebra Spherical Function Maximal Ideal Space## Preview

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