WIENER TAUBERIAN THEOREMS FOR VECTOR-VALUED FUNCTIONS

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. A weak form of Wiener’s Tauberian property is introduced and it is proved that LI(G,A) is weakly Tauberian if and only if A is. The vector analogue of Wiener’s L-span of translates theorem is examined.


INTRODUCTION.
Weiner's Tauberian theorem for the group algebra LI(G) of a locally compact abelian group G can be formulated in several ways. Here are two of them (see [1]): (I) For every proper closed ideal I in LI(G), its hull h(I) {7 E F: ?(7) 0 for every f in I} 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, then the closed ideal generated by f is the whole of LI(G).
What happens when we consider the "generalized" group algebra La(G,A) of A-valued integrable functions on G? ( (II') If f in L'(G) has non-vanishing Fourier transform, then the translation-invariant closed linear subspace spanned by f is the whole of LI(G).
The L analogue of this span-of-translates theorem is true and is also due to Wiener (see [3]).
In the last section of this note, we discuss the vector analogue of this L theorem.
Throughout A will denote a commutative, semisimple, regular Banach algebra. Th,'n LI(G,A) is also such an algebra ( [2], [4]). The analogues of (I) and (II) hold for L'(G,A)if we use Gelfand transforms provided A is Tauberian ( [2]). In this section we look at the situation when we consider A-valued Fourier transforms: The following natural analogue of (II), however, holds. THEOREM 2.3. Suppose that A has an identity. Let F LI(G,A). For the closed ideal generated by F to be the whole of LI(G,A), it is necessary and sufficient that qF(7) is invertible in A for each 7 F. PROOF. Since LI(G) has an approximate identity consisting of functions with compactly supported Fourier transforms, and since A has an identity, it follows that functions in La(G,A) with compactly supported Gelfand transforms are dense (that is, L(G,A) is "Tauberian").
Hence F generates La(G,A) as a closed ideal iff F has non-vanishing Gelfand transform iff (5F(7)) 0 for every complex homomorphism of A and 3' F WIENER TAUBERIAN THEOREMS FOR VECTOR-VALUED FUNCTIONS 477 iff ffF(3') is invertible for each -F.

WEAK TAUBERIAN PROPERTY.
Recall that A is Tauberian if given a in A and e > 0 there exits b in A with supp b coi)a(-t such that a-b < e. We weaken this condition and give th following definition. DEFINITION   To consider the vector analogue, we use the Plancherel theorem for vector valued functions due to Haussman [6]. The setting is s follows" G is -finite and A is a separable Hilbert space with a fixed orthonormal basis {e,} (with co-ordinatewise multiplication, A is a commutative semisimple Banach algebra with countable discrete maximal ideal space. But this is not important for our present purpose).
It turns out that only one part of the natural analogue of Wiener's theorem holds in the vector ce. Suppose that some f vanishes on a set E of finite positive measure in F. For ech n let E, {V e E: f,(v) 0} and let @. -,. Let @ (,) be the A-valued function on F with co-ordinate functions Since on S E, @ is a nonzero function in L(F,A). By elaucherel's threm ( [6], Theorem 4.4 and Example 5.3) @ F0 for a nonzero Fo in L(G,A). From the fact that f,@, 0 for each n, it follows, using Parseval's formula ( [6], Theorem 4.4), that F 0 is orthogonal to every translate of F. Thus the line span of the translates of F is not dense. MA 4.2. That the converse in not true can be seen follows. Chse , in L(F) with V real, non-vishing a.e. and #0. Define V,=$-V, =@, = -2 and @,=0 for n 3. Let F,F o be in L(G,A) with F (V,) and F 0 (@,). Then each co-ordinate function of F is nonvanishing a.e. but a simple computation shows that the nonzero element F 0 is orthogonal to every translate of F.