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Algebraic and Strong Splittings of Extensions of Banach Algebras
W. G. Bade, University of California, Berkeley, CA, H. G. Dales, University of Leeds, UK, and Z. A. Lykova, University of Newcastle, Newcastle Upon Tyne, UK
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Memoirs of the American Mathematical Society
1999; 113 pp; softcover
Volume: 137
ISBN-10: 0-8218-1058-8
ISBN-13: 978-0-8218-1058-3
List Price: US$50 Individual Members: US$30
Institutional Members: US\$40
Order Code: MEMO/137/656

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 $$A$$.

Graduate students and research mathematicians working in functional analysis.