Spectral synthesis and topologies on ideal spaces for Banach *-algebras

Feinstein, Joel; Kaniuth, E.; Somerset, D.W.B.

Abstract

This paper continues the study of spectral synthesis and the topologies τ∞ and τr on the ideal space of a Banach algebra, concentrating on the class of Banach *-algebras, and in particular on L1-group algebras. It is shown that if a group G is a finite extension of an abelian group then τr is Hausdorff on the ideal space of L1(G) if and only if L1(G) has spectral synthesis, which in turn is equivalent to G being compact. The result is applied to nilpotent groups, [FD]−-groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which L1(G) has spectral synthesis. It is also shown that if G is a non-discrete group then τr is not Hausdorff on the ideal lattice of the Fourier algebra A(G).

Citation

Feinstein, J., Kaniuth, E., & Somerset, D. (2002). Spectral synthesis and topologies on ideal spaces for Banach *-algebras. Journal of Functional Analysis, 196(1), doi:10.1006/jfan.2002.3964

• Journal Article Type: Article
• Acceptance Date: Feb 26, 2002
• Publication Date: Dec 1, 2002
• Deposit Date: Jul 30, 2001
• Publicly Available Date: Oct 9, 2007
• Journal: Journal of Functional Analysis
• Print ISSN: 0022-1236
• Electronic ISSN: 0022-1236
• Publisher: Elsevier
• Peer Reviewed: Not Peer Reviewed
• Volume: 196
• Issue: 1
• DOI: https://doi.org/10.1006/jfan.2002.3964
• Public URL: http://eprints.nottingham.ac.uk/id/eprint/15
• Publisher URL: http://www.sciencedirect.com/science/article/pii/S0022123602939649
• Copyright Statement: Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf