Noncommutative Geometry [Hardcover]

The main idea of this book is to generalize measure and operator theory to non-commutative situations. In the usual operator theory, von Neumann algebras serve as a generalization of "classical" measure theory. Commutative von Neumann algebras, or W*-algebras as they are sometimes called, are essentially bounded meausurable functions, and have measure spaces as their dual. These facts and a fine movtivation for the subject appear in the introduction to the book. The author shows with great clarity what is involved in extending measure theory to the non-commutative case. What is most interesting about the extensions is that they involve ideas from quantum physics. In addition, readers familiar with K-theory will see some brilliant uses of it in the book, particularly in the extension of BDF-theory to noncommutative situations, namely the KK-theory of Kasparov. The author also gives a taste of physics applications in the very last section of the book. He shows, interestingly, that when space-time is replaced by a product with a certain finite space, the Lagrangian of quantum electrodynamics becomes that of the Standard Model. Although such "add-ons" to space-time are not uncommon in physics (Kaluza-Klein theories being one example), the author's strategy is unique in its use of bimodules, and gives the three lepton generations.

There are also many other interesting topics as well in the book, such as how to deal with non-Hausdorff quotient spaces using noncommutative C*-algebras, deformation theory and the Kasparov group, the notion of Morita equivalence, leaf spaces of foliations, the E-theory of morphisms of separable C*-algebras, the extension of de Rham cohomology to a noncommutative framework (cyclic cohomology) and its relation to K-theory, the noncommutative torus and the quantum Hall effect.

The book is an excellent source of information on noncommutative geoemtry and with the many references given one can find more detailed proofs. It is a subject that will no doubt continue to make its presence known in mathematics (and physics) in years to come.