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 \(A\).