Abstract
Different versions of Wiener′s Tauberian theorem are discussed for the generalized group algebra L1(G, A) (of integrable functions on a locally compact abelian group G taking values in a commutative semisimple regular Banach algebra A) using A‐valued Fourier transforms. A weak form of Wiener′s Tauberian property is introduced and it is proved that L1(G, A) is weakly Tauberian if and only if A is. The vector analogue of Wiener′s L2‐span of translates theorem is examined.
Highlights
Weiner’s Tauberian theorem for the group algebra LI(G) of a locally compact abelian group G can be formulated in several ways
{7 I} h(I) E F: ?(7) 0 for every f in is nonempty. (Here F is the dual group of G and f is the Fourier transforms of f). (II) If a function f in LI(G) has non-vanishing Fourier transform, the closed ideal generated by f is the whole of LI(G)
If we consider the Gelfand transform, the Tauberian theorem holds for LI(G,A) provided it holds for A ([2])
Summary
Weiner’s Tauberian theorem for the group algebra LI(G) of a locally compact abelian group G can be formulated in several ways. Different versions of Wiener’s Tauberian theorem are discussed for the generalized group algebra LI(G,A) (of integrable functions on a locally compact abelian group G taking values in a commutative semisimple regular Banach algebra A) using A-valued Fourier transforms. (II) If a function f in LI(G) has non-vanishing Fourier transform, the closed ideal generated by f is the whole of LI(G).
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