Algebraic and Strong Splittings of Extensions of Banach Algebras (Memoirs of the American Mathematical Society No. 656)
By: Z. A. Lykova (author), etc. (author), H. G. Dales (author), W.G. Bade (author)Paperback
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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) Wedderbum 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 $A$.
Introduction The role of second cohomology groups From algebraic splittings to strong splittings Finite-dimensional extensions Algebraic and strong splittings of finite-dimensional extensions Summary References.
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- ID: 9780821810583
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