Algebraic and strong splittings of extensions of Banach algebras
About this Title
W. G. Bade, H. G. Dales and Z. A. Lykova
Publication: Memoirs of the American Mathematical Society
Publication Year: 1999; Volume 137, Number 656
ISBNs: 978-0-8218-1058-3 (print); 978-1-4704-0245-7 (online)
MathSciNet review: 1491607
MSC: Primary 46Hxx; Secondary 16E40, 46J45, 46L35
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.
Table of Contents
- 1. Introduction
- 2. The role of second cohomology groups
- 3. From algebraic splittings to strong splittings
- 4. Finite-dimensional extensions
- 5. Algebraic and strong splittings of finite-dimensional extensions
- 6. Summary