Some questions of spectral synthesis on spheres

This paper considers the Banach algebra $L^1(R^n)$ with the usual norm and convolution as multiplication. A characterization is given for closed ideals of $L^1(R^n)$ which are rotation invariant and have $S^{n-1}$ as spectrum, in terms of annihilators of certain collections of pseudomeasures. The main result of the paper is connected with a construction which yields an uncountable chain of closed ideals intermediate between neighboring invariant closed ideals with spectrum $S^{n-1}$. This construction associates an ideal $I(E)$ with a closed subset $E\subset S^{n-1}$. It is shown that if $\operatorname{int}E_1\neq\operatorname{int}E_2$ then $I(E_1)\neq I(E_2)$. Another result is the lack of a continuous projection from the largest to the smallest ideal when $n =3$, and when $n>3$, from an invariant ideal onto the neighboring smaller invariant ideal. A certain algebra of functions on the sphere which arises naturally in the construction of the intermediate ideals is also studied.
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