ABSTRACT

Let A be a Banach algebra, and let J 3 : 0 — / — • 2 1 — ^ 4 — O b e a n extension

of A, where 21 is a Banach algebra and / is a closed ideal in 21. The extension splits

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

tinuous homomorphism) 0 : A —• 21 such that TT O 0 is the identity on A.

We consider first for which Banach algebras A it is true that every extension of A

in a particular class of extensions splits, either algebraically or strongly, and second for

which Banach algebras it is true that every extension of A in a particular class which

splits algebraically also splits strongly.

These questions are closely related to the question when the algebra 21 has a (strong)

Wedderburn decomposition. The main technique for resolving these questions involves the

Banach cohomology group 1~t2(A,E) for a Banach .A-bimodule E, and related cohomol-

ogy groups.

Later chapters are particularly concerned with the case where the ideal / is finite-

dimensional.

We obtain results for many of the standard Banach algebras A.

Keywords: Banach algebra, extensions, Wedderburn decomposition, Hochschild co-

homology, finite-dimensional extensions, tensor algebra, derivation, point derivation, in-

tertwining map, automatic continuity, strong Ditkin algebra, C* -algebra, group algebra,

convolution algebra, continuously different iable functions, Beurling algebras, formal power

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