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-