Harmonic analysis of spherical functions on \(SU(1,1)\) (Q1203381)

Denote by \(L^ 1(K\backslash G/K)\) the algebra of spherical integrable functions on \(SU(1,1)\), with convolution as multiplication. This is a commutative semi-simple algebra, and we use its Gelfand transform to study the ideals in \(L^ 1(K\backslash G/K)\). In particular, we are interested in conditions on an ideal that ensure that it is all of \(L^ 1(K\backslash G/K)\), or that it is \(L_ 0^ 1(K\backslash G/K)\). Spherical functions on \(SU(1,1)\) are naturally represented as radial functions on the unit disk \(D\) in the complex plane. Using this representation, these results are applied to characterize harmonic and holomorphic functions on \(D\).