Different versions of Wiener's Tauberian theorem are discussed for the generalized group algebra L 1 ( 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 L 1 ( G , A ) is weakly Tauberian if and only if A is. The vector analogue of Wiener's L 2 -span of translates theorem is examined.