Algebras of Linear Transformations

by Douglas R. Farenick

Contents
• Linear Algebra
  • Vector Spaces and Duality
  • Direct Sums and Quotients
  • Inner-Product Spaces
  • The Spectral Theorem
  • Fields and Field Extensions
  • Existence of Bases for Infinite-Dimensional Spaces
• Algebras
  • Algebraic Structures
  • Algebras with a Prescribed Basis
  • Algebras of Linear Transformations
  • Inversion and Spectra
  • Division Algebras and Other Simple Algebras
• Invariant Subspaces
  • The Invariant-Subspace Lattice
  • Idempotents and Projections
  • Existence of Invariant Subspaces
  • Representations and Left Ideals
  • Functional Calculus and Polar Decomposition
• Semisimple Algebras
  • Nilpotent Algebras and the Nil Radical
  • Structure of Semisimple Algebras
  • Structure of Simple Algebras
  • Isomorphism Classes of Semisimple Algebras
• Operator Algebras
  • Von Neumann Algebras
  • Real and Complex Involutive Algebras
  • Representation of Operator Algebras
  • Wedderburn Theorems for Operator Algebras
  • C*-Algebras
• Tensor Products
  • Free Vector Spaces
  • Tensor Products of Vector Spaces
  • Tensor Products of Algebras
  • Tensor Products of Operator Algebras

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