Electronic ISBN:  9781470402457 
Product Code:  MEMO/137/656.E 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 137; 1999; 113 ppMSC: Primary 46; Secondary 16;
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 finitedimensional. Results are obtained for many of the standard Banach algebras \(A\).ReadershipGraduate students and research mathematicians working in functional analysis.

Table of Contents

Chapters

1. Introduction

2. The role of second cohomology groups

3. From algebraic splittings to strong splittings

4. Finitedimensional extensions

5. Algebraic and strong splittings of finitedimensional extensions

6. Summary


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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) 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 finitedimensional. Results are obtained for many of the standard Banach algebras \(A\).
Graduate students and research mathematicians working in functional analysis.

Chapters

1. Introduction

2. The role of second cohomology groups

3. From algebraic splittings to strong splittings

4. Finitedimensional extensions

5. Algebraic and strong splittings of finitedimensional extensions

6. Summary