Algebraic and Strong Splittings of Extensions of Banach Algebras (Memoirs of the American Mathematical Society) by W. G. Bade (PDF)

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In this volume, the authors address the following: Let $A$ be a Banach algebra, and let $sum: 0rightarrow Irightarrowmathfrak Aoversetpitolongrightarrow Arightarrow 0$ 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) $theta: Arightarrowmathfrak A$ such that $picirctheta$ 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 $A$.

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