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Denote by $L^1(K\setminus G/K)$ 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 $L^1(K\setminus G/K)$. In particular, we are interested in conditions on an ideal that ensure that it is all of $L^1(K\setminus G/K)$, or that it is $L_0^1(K\setminus G/K)$. 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...