Vol. 30, No. 1, 1969

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Ideal structure in generalized group algebras

Kjeld Laursen

Vol. 30 (1969), No. 1, 155–174

We study the algebra L1(G,A) of Bochner-integrable functions from a locally compact topological group G to a Banach algebra A. First we characterize closed ideals in L1(G,A) as subspaces that are translation invariant in a certain sense (Theorem 2.2). After that we establish some generalizations of Wiener’s tauberian theorem. The class of algebras under consideration consists of strongly semi-simple and completely regular Banach algebras. After this, in §3, we deal with spectral synthesis. Our main result (Corollary 3.6) states that if A does not admit spectral synthesis then neither does L1(G,A). In §4 we apply the theory of completely regular, strongly semi-simple Banach algebras to obtain some conditions sufficient to ensure that a given ideal is the intersection of the maximal regular ideals containing it.

Mathematical Subject Classification
Primary: 46.80
Secondary: 22.00
Received: 8 June 1968
Revised: 18 March 1969
Published: 1 July 1969
Kjeld Laursen