Basic Noncommutative Geometry by Masoud Khalkhali

By Masoud Khalkhali

This article presents an advent to noncommutative geometry and a few of its functions. it may be used both as a textbook for a graduate direction or for self-study. it will likely be helpful for graduate scholars and researchers in arithmetic and theoretical physics and all those people who are attracted to gaining an figuring out of the topic. One characteristic of this publication is the wealth of examples and workouts that support the reader to navigate in the course of the topic. whereas heritage fabric is supplied within the textual content and in different appendices, a few familiarity with easy notions of sensible research, algebraic topology, differential geometry and homological algebra at a primary yr graduate point is helpful.

Developed by way of Alain Connes because the past due Seventies, noncommutative geometry has came across many purposes to long-standing conjectures in topology and geometry and has lately made headways in theoretical physics and quantity conception. The booklet starts off with an in depth description of a few of the main pertinent algebrageometry correspondences by means of casting geometric notions in algebraic phrases, then proceeds within the moment bankruptcy to the assumption of a noncommutative area and the way it really is developed. The final chapters take care of homological instruments: cyclic cohomology and Connes–Chern characters in K-theory and K-homology, culminating in a single commutative diagram expressing the equality of topological and analytic index in a noncommutative atmosphere. functions to integrality of noncommutative topological invariants are given as well.

Two new sections were additional to this moment version: one issues the Gauss–Bonnet theorem and the definition and computation of the scalar curvature of the curved noncommutative torus, and the second one is a quick advent to Hopf cyclic cohomology. The bibliography has been prolonged and a few new examples are provided.

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Example text

Now, given a set S let B D 2S D ff W S ! 2g; where 2 ´ f0; 1g. Note that B is a complete atomic Boolean algebra. Any map f W S ! g/ ´ g B f , and S Ý 2S is a contravariant functor from the category of sets to the category of complete atomic Boolean algebras. B; 2/; where we now think of 2 as a Boolean algebra with two elements. Any algebra map f W B ! C induces a set map fO W Cy ! By by fO. / D B f for all 2 Cy . It 30 1 Examples of algebra-geometry correspondences can be shown that the two functors that we have just defined are anti-equivalences of categories, quasi-inverse to each other.

D B f for all 2 Cy . It 30 1 Examples of algebra-geometry correspondences can be shown that the two functors that we have just defined are anti-equivalences of categories, quasi-inverse to each other. Thus once again we have a duality between a certain category of geometric objects, namely sets, and a category of commutative algebras, namely complete atomic Boolean algebras. This result is a special case of the Stone duality between Boolean algebras and a certain class of topological spaces [111].

Then C € (D Cr €) is a unital commutative C -algebra and so by the Gelfand–Naimark theorem it is isomorphic to the algebra of continuous functions on a compact Hausdorff space X . €; T / be the group of unitary characters of directly in terms of €. Let € €, also known as the Pontryagin dual of €. Under pointwise multiplication and with the compact-open topology, it is a compact topological group and it is easy to see that it is homeomorphic to the space of characters of the commutative C -algebra C €.

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