Before continuing our own work in this memoir, we take this opportunity of summa-
rizing some earlier results on the splitting of sequences and the decomposition of algebras.
The earliest algebraic theory is due to Wedderburn ([We]). In a terminology of Ka-
plansky (see [Ja2]), an algebra A is an SBI-algebra if, for each a rad A, there exists
b £ rad A with 2b —b2 = a and such that {b}c = {a}c, where {a}c = {x A : ax = xa).
The proof of Wedderburn^ principal theorem establishes the following result.
1.10. THEOREM. Let A be an SBI-aigebra such that A/rad A is finite-dimensional.
Then A has a Wedderburn decomposition. D
An algebra whose radical is a nil ideal is an SBI-algebra, and so the above theorem
applies to finite-dimensional algebras A. (An early proof of this form of the theorem is
given in [Alb, Theorem 23].) It was noted by Feldman ([Fe]) that, since each Banach
algebra is an SBI-algebra, every Banach algebra such that A/ rad A is finite-dimensional
has a strong Wedderburn decomposition (see [Pa2, 8.1.4]). Feldman also gave the example
which we noted above of a commutative Banach algebra with one-dimensional radical, but
with no strong Wedderburn decomposition. It was shown in [BC1] that Feldman's example
has a Wedderburn decomposition. Further examples related to Feldman's example are
given in [Yo].
Throughout, we denote by C(Q) the algebra of all continuous, complex-valued func-
tions on a compact (Hausdorff) space ft; C(ft) is a commutative, unital Banach algebra
with respect to the uniform norm defined by
| / |
= sup{|/(x)| : x e Q} ( / C ( n ) ) .
We now consider Banach algebras 21 with the special property that 21/91 = C($a) ,
where 9t = rad 21. The basic results in the case where $21 is totally disconnected were
obtained by Bade and Curtis in [BC2].
1.11. THEOREM. Let 21 be a commutative Banach algebra with radical 91 such that
21/91 = C ( $ B ) where $21 is totally disconnected. Then the following conditions on 21
are equivalent:
(a) the set of idempotents in 21 is bounded;
(b) 21 has a Wedderburn decomposition;
(c) 21 has a strong Wedderburn decomposition.
In the case where the equivalent conditions are satisfied, the strong Wedderburn de-
composition is unique, and the complementary closed subalgebra to 91 is the closed linear
span of the set of idempotents of 21. The conditions are always satisfied in the case where
the idempotents in 21 satisfy a certain interpolation condition; this occurs, for example,
when $21 is extremely disconnected. The conditions are also satisfied in the case where
$21 is totally disconnected and 91 is nilpotent; this latter result was extended by Gorin
and Lin ([GoL]) to the case where 91 is topologically nilpotent. However an example is
Previous Page Next Page