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.
