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
vm
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