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Field name	Details
Dewey Class	512.5
Title	Algebras of Linear Transformations ([EBook] /) / by Douglas R. Farenick.
Author	Farenick, Douglas R.
Other name(s)	SpringerLink (Online service)
Publication	New York, NY : : Springer New York, , 2001.
Physical Details	XIV, 240 p. : online resource.
Series	Universitext 0172-5939
ISBN	9781461300977
Summary Note	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.:
Contents note	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.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-1-4613-0097-7
Links to Related Works	Subject References: Algebra . Linear and Multilinear Algebras, Matrix Theory . Mathematics . Matrix theory . Authors: Farenick, Douglas R. . Corporate Authors: SpringerLink (Online service) . Series: Universitext . Classification: 512.5 .

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Algebras of Linear Transformations

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