Spectral Synthesis and Topologies on Ideal Spaces for Banach*-Algebras

✍ Scribed by J.F. Feinstein; E. Kaniuth; D.W.B. Somerset

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 225 KB
• Volume: 196
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2002.3964

✦ Synopsis

This paper continues the study of spectral synthesis and the topologies t 1 and t r on the ideal space of a Banach algebra, concentrating on the class of Banach * -algebras, and in particular on L 1 -group algebras. It is shown that if a group G is a finite extension of an abelian group then t r is Hausdorff on the ideal space of L 1 ðGÞ if and only if L 1 ðGÞ has spectral synthesis, which in turn is equivalent to G being compact. The result is applied to nilpotent groups, ½FD À -groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which L 1 ðGÞ has spectral synthesis. It is also shown that if G is a non-discrete group then t r is not Hausdorff on the ideal lattice of the Fourier algebra AðGÞ: # 2002 Elsevier Science (USA)