# Algebraic and Strong Splittings of Extensions of Banach Algebras

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*W. G. Bade; H. G. Dales; Z. A. Lykova*

In this volume, the authors address the following:

Let \(A\) be a Banach algebra, and let \(\sum\:\ 0\rightarrow
I\rightarrow\mathfrak A\overset\pi\to\longrightarrow A\rightarrow 0\) be an extension of
\(A\), where \(\mathfrak A\) is a Banach algebra and \(I\) is
a closed ideal in \(\mathfrak A\). The extension splits algebraically
(respectively, splits strongly) if there is a homomorphism (respectively,
continuous homomorphism) \(\theta\: A\rightarrow\mathfrak A\) such that
\(\pi\circ\theta\) is the identity on \(A\).

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 \(\mathfrak
A\) has a (strong) Wedderburn decomposition. The main technique for
resolving these questions involves the Banach cohomology group \(\mathcal
H^2(A,E)\) for a Banach \(A\)-bimodule \(E\), and related
cohomology groups.

Later chapters are particularly concerned with the case where the ideal
\(I\) is finite-dimensional. Results are obtained for many of the
standard Banach algebras \(A\).