Introduction
Operator algebras are self-adjoint (closed under taking Hermitian adjoints), topologically closed algebras of bounded linear operators acting on Hilbert spaces, forming a central area of functional analysis that originated in the 1930s through the pioneering work of John von Neumann and Israel Gelfand. The field encompasses two principal classes: C*-algebras, which are closed in the norm topology and abstractly axiomatized by conditions relating the norm, involution, and multiplication, and von Neumann algebras (originally called “rings of operators”), which are C*-algebras additionally closed in the weak or strong operator topology and contain the identity operator. Operator algebras provide a powerful framework for studying arbitrary sets of operators simultaneously, generalizing spectral theory from single operators to operator systems, and they possess rich connections to diverse mathematical areas including algebraic topology, differential geometry, harmonic analysis, quantum field theory, and quantum computing. The theory is built upon understanding the noncommutative order structure imposed by positive operators and different operator topologies (norm, strong, and weak), with foundational results such as von Neumann’s Double Commutant Theorem and the Gelfand-Naimark theorems establishing the deep relationship between abstract algebraic structures and concrete operator realizations.
Contents
C*-Algebras and Gelfand Theory
Normed and Banach Algebras
Gelfand Representation
C*-Algebras
The Gelfand Theorem for Commutative C*-Algebras
Positive Elements and Ideals in C*-Algebras
The Gelfand-Naimark Theorem
Toeplitz Operators
Fredholm Operators
C*-Dynamical Systems
C*-Dynamical Systems
Crossed Products of C*-Algebras
C*-Dynamical Systems with Integer Actions
Von Neumann Algebras
Von Neumann Algebras
The Double Commutant Theorem
Factors and Type Classification
Tomita-Takesaki Theory