On the spectral synthesis problem for points in the dual of a nilpotent Lie group
Let us denote by \({\mathcal I}\) the space of two-sided closed ideals of a semisimple Banach algebra A with involution, and by \(\Pr im_*(A)\) the space of kernels of the topologically irreducible unitary representations of A. If I is a two-sided ideal of A, its hull \(h(I)\subseteq \Pr im_*(A)\) is the set of primitive ideals containing I. On the other hand, if C is a closed subset of \(\Pr im_*(A)\) (in the Jacobson topology), then its kernel \({\mathcal I}_ C\subseteq {\mathcal I}\) is the set of two-sided closed ideals whose hull is C. A subset C of \(\Pr im_*(A)\) is a set of spectral synthesis if its kernel consists only of the ideal \(\text{ker}C = \cap \{J:J\in C\}\). If G is a connected, simply connected nilpotent Lie group, then every point in \(\Pr im_*(L^ 1(G))\) is a set of spectral synthesis provided that the nilpotency length of G is not larger than 2. This paper shows that in general points in \(\Pr im_*(L^ 1(G))\) are not sets of synthesis if the length of G is 3. The proof is based upon the realization of closed subsets of \(\Pr im_*(L^ 1(G))\) as closed subsets of the dual space \({\mathfrak g}^*\) of the Lie algebra of G which are invariant under the coadjoint action of G, along the lines of the Kirillov's orbit theory.
kernels of the topologically irreducible unitary representations
connected, simply connected nilpotent Lie group
scientific article; zbMATH DE number 3849826