A century after physicists began to realize that classical mechanics is wrong in the atomic realm, and so cannot be a correct description of nature, Weaver (mathematics, Washington U., St. Louis, Missouri) declares it is now possible to construct a simple, unified approach to the general concept of quantization in mathematics, then gets on with it. He provides analogies between classical concepts and their quantum equivalents, among them propositional logic and quantum logic, entire functions and Bargmann-Segal multipliers, and metric spaces and Lipschitz algebras. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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.
