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More About This Textbook
Overview
With a unique approach and presenting an array of new and intriguing topics, Mathematical Quantization offers a survey of operator algebras and related structures from the point of view that these objects are quantizations of classical mathematical structures. This approach makes possible, with minimal mathematical detail, a unified treatment of a variety of topics.
Detailed here for the first time, the fundamental idea of mathematical quantization is that sets are replaced by Hilbert spaces. Building on this idea, and most importantly on the fact that scalar-valued functions on a set correspond to operators on a Hilbert space, one can determine quantum analogs of a variety of classical structures. In particular, because topologies and measure classes on a set can be treated in terms of scalar-valued functions, we can transfer these constructions to the quantum realm, giving rise to C*- and von Neumann algebras.
In the first half of the book, the author quickly builds the operator algebra setting. He uses this as a unifying theme in the second half, in which he treats several active research topics, some for the first time in book form. These include the quantum plane and tori, operator spaces, Hilbert modules, Lipschitz algebras, and quantum groups.
For graduate students, Mathematical Quantization offers an ideal introduction to a research area of great current interest. For professionals in operator algebras and functional analysis, it provides a readable tour of the current state of the field.
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Table of Contents
QUANTUM MECHANICS
Classical Physics
States and Events
Observables
Dynamics
Composite Systems
Quantum Computation
HILBERT SPACES
Definitions and Examples
Subspaces
Orthonormal Bases
Duals and Direct Sums
Tensor Products
Quantum Logic
OPERATORS
Unitaries and Projections
Continuous Functional Calculus
Borel Functional Calculus
Spectral Measures
The Bounded Spectral Theorem
Unbounded Operators
The Unbounded Spectral Theorem
Stone's Theorem
THE QUANTUM PLANE
Position and Momentum
The Tracial Representation
Bargmann-Segal Space
Quantum Complex Analysis
C*-ALGEBRAS
The Algebras C(X)
Topologies from Functions
Abelian C*-Algebras
The Quantum Plane
Quantum Tori
The GNS Construction
VON NEUMANN ALGEBRAS
The Algebras l8 (X)
The Algebras L8 (X)
Trace Class Operators
The Algebras B(H)
Von Neumann Algebras
The Quantum Plane and Tori
QUANTUM FIELD THEORY
Fock Space
CCR Algebras
Realtivistic Particles
Flat Spacetime
Curved Spacetime
OPERATOR SPACES
The Spaces V(K)
Mstiex Norms and Convexity
Duality
Matrix-Valued Functions
Operator Systems
HILBERT MODULES
Continuous Hilbert Bundles
Hilbert L8-Modules
Hilber C*-Modules
Hilbert W*-Modules
Crossed Products
Hilbert *-Bimodules
LIPSCHITZ ALGEBRAS
The Algebras Lip0(X)
Measurable Metrics
The Derivation Theorem
Examples
Quantum Markov Semigroups
QUANTUM GROUPS
Finite Dimensional C*-Algebras
Finite Quantum Groups
Compact Quantum Groups
Haar Measure\
REFERENCES