Mathematical Quantization

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.

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