16
W. G. BADE, H. G. DALES, Z. A. LYKOVA
of $A The ideal which is maximal among those with hull equal to E is
I(E) = {feA:f\E = 0},
and the ideal which is minimal among those with hull equal to E is
J(E) = {/ A : supp / is compact and (supp / ) Pi E 0} .
The set E is a set of non-synthesis if J(E) ^ 1(E). In this case, set 21 = A/J(E). Then
rad2l = 1(E)/1(E) and 2l/rad2l = A/J(£ ) = A(E). One can ask whether 21 has a
(strong) Wedderburn decomposition. Consider the case where A = Ll(G) is the Fourier
algebra of a locally compact group G (so that $A = T, the dual group of G), and E is
a set of non-synthesis for A. It was proved by Bachelis and Saeki ([BS]) that 21 does not
have a strong Wedderburn decomposition; this result was extended in [BDa2] to show that
21 does not have any Wedderburn decomposition. The same result is proved for certain
Beurling algebras. A weight function on Rk is a continuous function u : R^ —• R + \ {0}
such that
uj(s + t)u(s)u;(t) (s,tGE f c ) ;
we then set
L^R^LU) = {/ C-valued, measurable on
Rk
: ||/||w = / \f(t)\u(t)dt oo} ,
so that L1(IR/c,a;) is a Beurling algebra with respect to the convolution product ^ on
Rk . For example, set
ua(t) = (l + \t\)a (teR)
where a 0. The algebra of Fourier transforms of L1(Rk,toa) is j4Q(Rfe), a Banach
function algebra on
Rk.
Let E be a set of non-synthesis for Aa. It is proved in [BDa2]
that, in the case where 0 a 1/2, the quotient Aa/J(E) does not have a Wedderburn
decomposition, but it is not known whether or not this result holds in the case where
1/2 a 1. For a generalization of the result in [BDa2], see [Whi, Corollary 3.7].
We note that it appears to be an open question whether or not rad 21 is complemented
as a Banach subspace of 21 in the case where 21 = A/J(E) and A = LX(G), in the above
notation.
We pointed out that an example of a non-admissible, singular extension of a (commu-
tative) semisimple Banach algebra has been constructed by Yakovlev ([Ya]). We wish to
conclude this introduction by describing this instructive and rather elementary example.
Let E be a linear space. Then the tensor algebra of E,
® « = ©{®^:i6Z+},
is well-known in multilinear algebra (see [Gre, Chapter 3], for example); the product in
(££) E is specified by
k i+j=k
The 'Banach' version of this algebra has also been studied.
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