Algebras of Linear Transformations / Edition 1

by Douglas R. Farenick

Overview

The aim of this book is twofold: (i) to give an exposition of the basic theory of finite-dimensional algebras at a levelthat isappropriate for senior undergraduate and first-year graduate students, and (ii) to provide the mathematical foundation needed to prepare the reader for the advanced study of anyone of several fields of mathematics. The subject under study is by no means new-indeed it is classical­ yet a book that offers a straightforward and concrete treatment of this theory seems justified for several reasons. First, algebras and linear transformations in one guise or another are standard features of various parts of modern mathematics. These include well-entrenched fields such as representation theory, as well as newer ones such as quantum groups. Second, a study of the elementary theory offinite-dimensional algebras is particularly useful in motivating and casting light upon more sophisticated topics such as module theory and operator algebras. Indeed, the reader who acquires a good understanding of the basic theory of algebras is wellpositioned to ap­ preciate results in operator algebras, representation theory, and ring theory. In return for their efforts, readers are rewarded by the results themselves, several of which are fundamental theorems of striking elegance.

Product Details

ISBN-13:	9780387950624
Publisher:	Springer New York
Publication date:	10/06/2000
Series:	Universitext
Edition description:	2001
Pages:	239
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

1. Linear Algebra.- 1.1 Vector Spaces and Duality.- 1.2 Direct Sums and Quotients.- 1.3 Inner-Product Spaces.- 1.4 The Spectral Theorem.- 1.5 Fields and Field Extensions.- 1.6 Existence of Bases for Infinite-Dimensional Spaces.- 1.7 Notes.- 1.8 Exercises.- 2. Algebras.- 2.1 Algebrai c Structures.- 2.2 Algebras with a Prescribed Basis.- 2.3 Algebras of Linear Transformations.- 2.4 Inversion and Spectra.- 2.5 Division Algebras and Other Simple Algebras.- 2.6 Notes.- 2.7 Exercises.- 3. Invariant Subspaces.- 3.1 The Invariant-Subspace Lattice.- 3.2 Idempotents and Projections.- 3.3 Existence of Invariant Subspaces.- 3.4 Representations and Left Ideals.- 3.5 Functional Calculus and Polar Decomposition.- 3.6 Notes.- 3.7 Exercises.- 4. Semisimple Algebras.- 4.1 Nilpotent Algebras and the Nil Radical.- 4.2 Structure of Semisimple Algebras.- 4.3 Structure of Simple Algebras.- 4.4 Isomorphism Classes of Semisimple Algebras.- 4.5 Notes.- 4.6 Exercises.- 5. Operator Algebras.- 5.1 Von Neumann Algebras.- 5.2 Real and Complex Involutive Algebras.- 5.3 Representation of Operator Algebras.- 5.4 Wedderburn Theorems for Operator Algebras.- 5.5 C*-Algebras.- 5.5 Notes.- 5.7 Exercises.- 6. Tensor Products.- 6.1 Free Vector Spaces.- 6.2 Tensor Products of Vector Spaces.- 6.3 Tensor Products of Algebras.- 6.4 Tensor Products of Operator Algebras.- 6.5 Notes.- 6.6 Exercises.- References.