1. INTRODUCTION

Let 21 be a finite-dimensional algebra (over C). Then the Wedderburn principal theo-

rem asserts that there is a subalgebra 93 of 21 such that 21 is the direct sum

21 = 03 0 rad 21,

where rad 21 is the radical of 21. Motivated by this theorem, many authors have studied

when an infinite-dimensional Banach algebra 21 has an analogous decomposition. Imme-

diately we see that we must distinguish the cases of a Wedderburn decomposition of 21,

where there is a subalgebra 03 of 21 with 21 = 03 0 rad 21, and of a strong Wedderburn

decomposition of 21, where there is a closed subalgebra 03 of 21 with this property.

The first context in which this question was studied was that in which 21 is a specified

non-semisimple Banach algebra. However, our main interest in the present work is to

regard the quotient algebra

,4 = 21/rad 21

as being specified, and to discuss questions of the existence of decompositions of the Banach

algebras 21 when 21 is an arbitrary member of a particular class of Banach algebras. In

fact we shall work in a more general situation; we shall consider extensions of a Banach

algebra A, where an extension of A is defined as a certain short exact sequence

(see Definition 1.2); the most important case is that where / = rad 21. The extension

splits algebraically (respectively, splits strongly) if there is a homomorphism (respectively,

a continuous homomorphism) 6 : A — 21 such that n o 0 = iA, the identity on A. We are

seeking to determine when each such extension J^ of A splits algebraically, when each

such extension splits strongly, and when the existence of an algebraic splitting implies that

there is a strong splitting; the latter question is an 'automatic continuity' question.

Our most extensive results apply to algebraic splittings of finite-dimensional extensions

of (especially commutative) Banach algebras; this question has not previously received

much attention. A prominent feature of this study is that certain reductions to the one-

dimensional case that apply in the case of strong splittings are no longer available, and so

we must engage directly with the general finite-dimensional case.

The main established technique that is used to consider splittings of extensions of a

Banach algebra A is to calculate H2(A,E), the second Banach cohomology group of A

with coefficients in a Banach v4-bimodule E. (This technique builds on an earlier algebraic

method of Hochschild.) We also consider a related cohomology group H2(A, E) (see §2).

In certain situations, this technique is very effective, but it does have limitations. First,

the algebraic theory only applies in the special case where the extension ^ is singular,

Manuscript received by the editor January 6, 1997

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