SPLITTINGS OF EXTENSIONS OF BANACH ALGEBRAS 3

nach algebra to split, either algebraically or strongly. For many standard examples of

Banach algebras, all finite-dimensional extensions split strongly, but we shall also give

various examples of finite-dimensional extensions which do not split even algebraically.

Two key theorems are Theorems 4.9 and 4.13, dealing with strong splittings and alge-

braic splittings, respectively. Special cases of these theorems assert the following. Let

A be a commutative, unital Banach algebra, and suppose that each maximal ideal of A

has a bounded approximate identity (respectively, an approximate identity). Then each

finite-dimensional extension of A splits strongly (respectively, splits algebraically). Chap-

ter 4 concludes with a further investigation of when all intertwining maps from Banach

algebras—now into finite-dimensional modules—are automatically continuous.

In Chapter 5, we shall apply the general results of Chapter 4 to some specific examples,

concentrating on the case of extensions of commutative Banach algebras. In the case of

strong splittings, the case of finite-dimensional extensions can often be reduced to that

of one-dimensional extensions, but this is not possible for algebraic extensions, and the

results may depend on the dimension of the extension. For example, we consider the

algebra C^n\l) of n -times continuously differentiable functions on I; it will be shown in

Theorem 5.7 that extensions of C^n\l) of dimension at most n split algebraically (but not

necessarily strongly), but that there is such an extension of dimension n + 1 which does

not split algebraically. We shall also investigate some Banach function algebras related to

C^(l). In Theorem 5.12, we shall prove that all finite-dimensional extensions of certain

local Banach algebras of power series split algebraically; the algebraic calculations for

this result are rather complicated because it again seems that there is no straightforward

reduction from the n-dimensional to the 1-dimensional case.

Finally, in Chapter 6, we shall summarize the results that we have obtained for various

classes of Banach algebras, and raise some open questions.

We are very grateful to Dr. Olaf Ermert for some valuable comments on an earlier

version of this memoir and to Dr. H. Steiniger and Dr. Y. Selivanov for some corrections.

This work was supported by three agencies. First, Z. A. Lykova was awarded a Royal

Society Fellowship to enable her to visit the University of Leeds to work with H. G. Dales

in the period March-June, 1993, and an RFFI grant 93-011-156; she thanks the School

of Mathematics at Leeds and the Department of Mathematics at Berkeley for hospitality

while this work was carried out. Second, W. G. Bade and H. G. Dales were awarded

Collaborative Research Grant No. 940050 by NATO, enabling them to visit each other.

They acknowledge with thanks this financial support.

We now give a fuller description of our main results, establish some preliminary no-

tations and conventions, give precise definitions for the questions that concern us, prove

some general results that will be used throughout the memoir, and summarize some earlier

results in this area.

Throughout this memoir we shall be concerned with extensions of a fixed Banach al-

gebra A; the following is a full definition of the context in which we shall work. For