Algebraic and Strong Splittings of Extensions of Banach Algebras (Memoirs of the American Mathematical Society No. 656)

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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Description

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 $A$.

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Contents

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.

Product Details

  • publication date: 15/01/1999
  • ISBN13: 9780821810583
  • Format: Paperback
  • Number Of Pages: 113
  • ID: 9780821810583
  • weight: 255
  • ISBN10: 0821810588

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