2
W. G. BADE, H. G. DALES, Z. A. LYKOVA
i.e., I2 = 0. (However, in certain cases, results obtained in the singular case lead to
results in the more general, nilpotent, case in which In 0 for some n N.) Second, the
cohomology theory as applied to Banach algebras requires the a priori assumption that
the short exact sequence J2 ^ e admissible; i.e., that the closed ideal / be complemented
in the Banach algebra 21 as a Banach space. In general, there is no reason for such a short
exact sequence to be admissible; we describe a counter-example in §1. However, in the case
where J is a finite-dimensional ideal in 21, the sequence J2 1S automatically admissible.
Further, we can reduce the case of the strong splitting of an arbitrary finite-dimensional
extension to that of singular, finite-dimensional extensions (see Theorem 1.8), and so the
cohomology theory gives the full story in the case of strong splittings of finite-dimensional
extensions; if A is commutative, we can even reduce to one-dimensional extensions (see
Theorem 4.4). In general, the question of the algebraic splitting of finite-dimensional
extensions of a Banach algebra is more difficult because there appears to be no reduction
to the singular case, but we can obtain such a reduction in the case of finite-dimensional
extensions of commutative Banach algebras. However, in this context, there is certainly
no reduction of the finite-dimensional, singular case to that of one-dimensional extensions.
The seminal results connecting the theory of Wedderburn decompositions of Banach
algebras with the second Banach cohomology groups were given by Professor H. Kamowitz
in 1962. Later the theory of Wedderburn decompositions of Banach algebras was further
developed somewhat independently by the Moscow school led by Professor A. Ya. Helem-
skii and by certain Western authors, particularly Professor B. E. Johnson of Newcastle,
England. (More details of the history of our subject are given later in this introduction.)
In fact, the existing results in the literature are rather scattered; to give a full picture of
the subject, we have tried to collect these scattered results for ease of future reference,
and occasionally we have given a proof of existing theorems.
The second part of this chapter introduces our notation and formulates more precisely
the questions which we wish to consider; it also gives some elementary reductions, and
summarizes some earlier work.
In Chapter 2, we shall define the cohomology groups
H2(A,E), H2(A,E),
and
H2(A,E), and explain their role in our theory. The question whether or not the fact
that H2(A, E) = {0} implies that H2(A,E) = {0} is related to the question of when
the existence of an algebraic splitting implies that there is a strong splitting; we define
intertwining maps, and draw attention to their significant role in the theory. It is a very
relevant question for us to determine when all intertwining maps from a Banach algebra
A into a Banach ^4-bimodule E are automatically continuous.
Chapter 3 is devoted to a discussion of the question when we can deduce the existence
of a strong splitting of an extension from that of an algebraic splitting; we shall give a
variety of examples. The results obtained allow us to exhibit extensions of many standard
Banach algebras such that the extension does not split even algebraically.
In Chapter 4, we shall concentrate on finite-dimensional extensions, and obtain some
theorems giving sufficient conditions for all finite-dimensional extensions of a given Ba-
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