Algebraic and strong splittings of extensions of Banach algebras / W.G. Bade, H.G. Dales, Z.A. Lykova.

Format:

Book

Author/Creator:

Badè, W. G. (William G.), 1924-2012, author.

Dales, H. G. (Harold G.), 1944- author.

Lykova, Z. A. (Zinaida Alexandrovna), 1954- author.

Series:

Memoirs of the American Mathematical Society ; no. 656.

Memoirs of the American Mathematical Society, 0065-9266 ; number 656

Language:

English

Subjects (All):

Banach algebras.

Ideals (Algebra).

Modules (Algebra).

Continuity.

Physical Description:

1 online resource (129 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1999]

Language Note:

English

Summary:

In this volume, the authors address the following: Let $A$ be a Banach algebra, and let $\sum\:\ 0\rightarrow I\rightarrow\frak A\overset\pi\to\longrightarrow A\rightarrow 0$ be an extension of $A$, where $\frak A$ is a Banach algebra and $I$ is a closed ideal in $\frak A$. The extension splits algebraically (respectively, splits strongly) if there is a homomorphism (respectively, continuous homomorphism) $\theta\: A\rightarrow\frak A$ 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 $\frak A$ has a (strong) Wedderburn decomposition. The main technique for resolving these questions involves the Banach cohomology group $\cal H2(A,E)$ 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 $A$.

Contents:

""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""; ""REFERENCES""

Notes:

"January 1999, volume 137, number 656 (fifth of 6 numbers)."

Includes bibliographical references (pages 107-113).

Description based on print version record.

ISBN:

1-4704-0245-9