Group Algebras and Applications
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 µr 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.
KeywordsBanach Algebra Heisenberg Group Group Algebra Spherical Function Maximal Ideal Space
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