Spectral synthesis and topologies on ideal spaces for Banach *-algebras
Feinstein, J. F. and Kaniuth, E. (1999) Spectral synthesis and topologies on ideal spaces for Banach *-algebras.
Abstract This paper continues the study of spectral synthesis and the topologies Tau-infinity and Tau-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 infnite extension of an abelian group then Tau-r is Hausdorff on the ideal space of L 1 (G) if and only if L 1 (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 L 1 (G) has spectral synthesis. It is also shown that if G is a non-discrete group then Tau-r is not Hausdorff on the ideal lattice of the Fourier algebra A(G).
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