Algebras of Linear Transformations

Front Cover
Springer Science & Business Media, Oct 6, 2000 - Mathematics - 240 pages
0 Reviews
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 trans formations in one guise or another are standard features of various parts of modern mathematics. These include well-entrenched fields such as repre sentation theory, as well as newer ones such as quantum groups. Second, a study ofthe 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.
 

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

Linear Algebra
xiii
12 Direct Sums and Quotients
6
13 InnerProduct Spaces
9
14 The Spectral Theorem
18
15 Fields and Field Extensions
24
16 Existence of Bases for InfiniteDimensional Spaces
27
17 Notes
30
18 Exercises
31
42 Structure of Semisimple Algebras
124
43 Structure of Simple Algebras
131
44 Isomorphism Classes of Semisimple Algebras
139
45 Notes
143
46 Exercises
144
Operator Algebras
149
52 Real and Complex Involutive Algebras
157
53 Representation of Operator Algebras
165

Algebras
37
22 Algebras with a Prescribed Basis
43
23 Algebras of Linear Transformations
47
24 Inversion and Spectra
54
25 Division Algebras and Other Simple Algebras
62
26 Notes
68
Invariant Subspaces
75
32 Idempotents and Projections
79
33 Existence of Invariant Subspaces
85
34 Representations and Left Ideals
94
35 Functional Calculus and Polar Decomposition
101
36 Notes
109
37 Exercises
110
Semisimple Algebras
115
54 Wedderburn Theorems for Operator Algebras
171
55 CAlgebras
176
56 Notes
184
57 Exercises
185
Tensor Products
191
62 Tensor Products of Vector Spaces
195
63 Tensor Products of Algebras
210
64 Tensor Products of Operator Algebras
220
65 Notes
225
66 Exercises
227
References
231
Index
233
Copyright

Other editions - View all

Common terms and phrases

Popular passages

Page ix - " This work has been the pleasantest mathematical effort of my life. In no other have I seemed to myself to have received so full a reward for my mental labor in the novelty and breadth of the results.
Page v - There is still something in the system that gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties...
Page v - In his terminology primitive algebra means the same thing as what we now call division algebra. This extraordinary result has excited the fantasy of every algebraist and still does so in our day. Very great efforts have been directed toward a deeper understanding of its meaning. In the first period following his discovery the work consisted mainly in a polishing up of his proofs. But the fundamental ideas of all these later proofs are already contained in his memoir. In the meantime a great change...

References to this book

Bibliographic information