On closed ideals in the motion group algebra

Abstract

A right closed ideal is said to be right-primary if it is contained in precisely one right maximal closed ideal. A set E C d//where E = h(k(E)), is said to be a set of spectral synthesis if I = k(E) is the only right closed ideal such that h(1)= E. For the Euclidean motion group M(2) we have proved in [4] that 0 is not a set of spectral synthesis. That is, the one-sided Wiener's Tauberian Theorem does not hold for M(2). Our main purpose is to generalize this result by showing that for M(2) many subsets of ~ fail to be sets of spectral synthesis. In particular, every member of J// contains infinitely many right-primary ideals, whereas in the abelian case every closed primary ideal is maximal [3, p. 326]. Using the result in [4] and his methods in [-1, 2], Leptin (in a letter to the author) obtained the same results for generalized Ll-algebras (see Appendix).