On the spectral synthesis problem for points in the dual of a nilpotent Lie group
type: Item
instance of: scholarly article
title: On the spectral synthesis problem for points in the dual of a nilpotent Lie group
zbMATH Open document ID: 0535.43004
DOI: 10.1007/BF02384304
author: Jean Ludwig
published in: Arkiv för Matematik
publication date: 1983-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 43A45
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 3849826
zbMATH Keywords: twosided closed ideals
zbMATH Keywords: kernels of the topologically irreducible unitary representations
zbMATH Keywords: Jacobson topology
zbMATH Keywords: set of spectral synthesis
zbMATH Keywords: connected, simply connected nilpotent Lie group
zbMATH Keywords: dual space
zbMATH Keywords: primitive ideal
zbMATH Keywords: hull of an ideal
zbMATH Keywords: coadjoint action
zbMATH Keywords: Kirillov's orbit theory
Wikidata QID: Q115392370
reviewed by: Massimo A. Picardello
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3849826