Spectral synthesis and topologies on ideal spaces for Banach *-algebras
Feinstein, Joel and Kaniuth, E. and Somerset, D.W.B. (2002) Spectral synthesis and topologies on ideal spaces for Banach *-algebras. Journal of Functional Analysis, 196 (1). pp. 19-39. ISSN 0022-1236
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).
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