Algebras of Linear Transformations

This book studies algebras and linear transformations acting on finite -dimensional vector spaces over arbitrary fields. It is written for students who have prior knowledge of algebra and linear algebra. The goal is to present a balance of theory and example in order for students to gain a firm understanding of the basic theory of finite- dimensional algebras and to provide a foundation for subsequent advanced study in a number of areas of mathematics. As such, the level of exposition is suitable for senior undergraduate students.

Inhalt

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.