An Introduction to Banach Algebras and Operator Algebras

This book serves as an introduction to Banach algebra theory, operator theory, and C*-algebras. It is aimed at graduate students with basic knowledge of functional analysis, or researchers who wish to pursue research in these areas.

While covering the standard material necessary to embark on any further exploration of the theory (Gelfand theory, the GNS construction and the spectral theorem for normal operators), the book also presents specialized material and topics of recent interest, including the holomorphic functional calculus and upper-semicontinuity of the spectrum, Jacobson's Density Theorem, the Cohen–Hewitt Factorisation Theorem, Kaplansky's Density Theorem and Kadison's Transitivity Theorem.

The presentation is detailed and clear, with special attention to highlighting the scope and limitations of the theory through examples. Each chapter includes Notes and Remarks which provide extra context, and in many instances relate the material to recent results in the research literature and to open problems.

The appendices are not normally found in introductory textbooks, which include Fredholm theory for Hilbert space operators, Arens regularity, the injective and projective tensor products of Banach spaces, the Schatten p-classes, and the Sherman–Takeda Theorem.

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• Banach Algebras, An Introduction
• The Holomorphic Functional Calculus
• The Spectrum
• Abelian Banach Algebras
• The Algebra of Banach Space Operators
• The Algebra of Hilbert Space Operators
• Representations of Banach Algebras
• C*-Algebras, An Introduction
• C*-Algebras, Approximate Identities and Ideals
• The GNS Construction
• von Neumann Algebras
• The Spectral Theorem for Normal Operators
• Appendices:
  • A Brief Review of Banach Space Theory
  • Fredholm Theory
  • B(H) as a Dual Space
  • Arens Products and the Sherman-Takeda Theorem
  • The Essential Spectrum of a Hilbert Space Operator