An Introduction to Operator Algebras is a concise text/reference that focuses on the fundamental results in operator algebras. Results discussed include Gelfand's representation of commutative C*-algebras, the GNS construction, the spectral theorem, polar decomposition, von Neumann's double commutant theorem, Kaplansky's density theorem, the (continuous, Borel, and L8) functional calculus for normal operators, and type decomposition for von Neumann algebras. Exercises are provided after each chapter.

part I|48 pages

Banach Algebras

chapter 1|6 pages

Review on Functional Analysis

chapter 2|7 pages

Banach Algebras and the Invertible Group

chapter 3|6 pages

The Spectrum

chapter 4|6 pages

Multiplicative Linear Functionals

chapter 5|7 pages

The Gelfand Transform and Applications

chapter 6|7 pages

Examples of Maximal Ideal Spaces

chapter 7|8 pages

Non-Unital Banach Algebras

part II|48 pages


chapter 8|5 pages


chapter 9|5 pages

Commutative C*-Algebras

chapter 10|6 pages

The Spectral Theorem and Applications

chapter 11|5 pages

Further Applications

chapter 12|6 pages

Polar Decomposition

chapter 13|7 pages

Positive Linear Functionals and States

chapter 14|5 pages

The GNS Construction

chapter 15|8 pages

Non-Unital C*-Algebras

part III|57 pages

Von Neumann Algebras

chapter 16|5 pages

Strong- and Weak-Operator Topologies

chapter 17|5 pages

Existence of Projections

chapter 18|5 pages

The Double Commutant Theorem

chapter 19|5 pages

The Kaplansky Density Theorem

chapter 20|6 pages

The Borel Functional Calculus

chapter 21|5 pages

L∞ as a von Neumann Algebra

chapter 22|5 pages

Abelian von Neumann Algebras

chapter 23|6 pages

The L∞-Functional Calculus

chapter 24|5 pages

Equivalence of Projections

chapter 25|4 pages

A Partial Ordering

chapter 26|5 pages

Type Decomposition