Algebras of Linear Transformations
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.
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12 Direct Sums and Quotients
13 InnerProduct Spaces
14 The Spectral Theorem
15 Fields and Field Extensions
16 Existence of Bases for InfiniteDimensional Spaces
42 Structure of Semisimple Algebras
43 Structure of Simple Algebras
44 Isomorphism Classes of Semisimple Algebras
52 Real and Complex Involutive Algebras
53 Representation of Operator Algebras
22 Algebras with a Prescribed Basis
23 Algebras of Linear Transformations
24 Inversion and Spectra
25 Division Algebras and Other Simple Algebras
32 Idempotents and Projections
33 Existence of Invariant Subspaces
34 Representations and Left Ideals
35 Functional Calculus and Polar Decomposition
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algebra 21 algebras of linear arbitrary Assume basis elements basis vectors bilinear Burnside's Theorem C*-algebra commutative complex inner-product space complex numbers complex operator algebra complexification consider decomposition defined denote dimension direct sum division algebra eigenvectors elementary tensors equation Example exist field F finite finite-dimensional algebra finite-dimensional complex finite-dimensional real finite-dimensional vector space group algebra Hence hermitian homomorphism ideal of 21 idempotent implies injective homomorphism inner-product space invariant subspace invertible involution involutive algebra irreducible representation isomorphism left ideal left regular representation Lemma linear algebra linear functional linear transformation linearly independent matrix representation minimal polynomial Mn(C Mn(F Neumann algebra norm operator algebra orthonormal basis Proof properly nilpotent Proposition Prove quotient real algebra real operator algebra scalar semisimple algebra simple algebra span Spectral Theorem Suppose tensor product theory unital algebra unital subalgebra unitary universal property upper-triangular Wedderburn zero
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 - 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...