An Introduction to Noncommutative Spaces and Their Geometry

By Landi, Giovanni

Book Id: WPLBN0000659371
Format Type: PDF eBook:
File Size: 1.39 MB
Reproduction Date: 2005

Title:	An Introduction to Noncommutative Spaces and Their Geometry
Author:	Landi, Giovanni
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Language:	English
Subject:	Science., Mathematics, Logic
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Citation APA MLA Chicago Landi, B. G. (n.d.). An Introduction to Noncommutative Spaces and Their Geometry. Retrieved from http://gutenberg.us/

Description
Mathematics document containing theorems and formulas.

Excerpt
Excerpt: These notes arose from a series of introductory seminars on noncommutative geometry I gave at the University of Trieste in September 1995 during the XWorkshop on Differential GeometricMethods in Classical Mechanics. It was Beppe Marmo's suggestion that I wrote notes of the lectures. The notes are mainly an introduction to Connes' noncommutative geometry. They could serve as a ` rst aid kit' before one ventures into the beautiful but bewildering landscape of Connes' theory [25]. The main di erence with other available introductions to Connes's work, notably Kastler's papers [65] and also Gracia-Bond a and Varilly paper [101], is the emphasis on noncommutative spaces seen as concrete spaces.

Table of Contents
Contents 1 Introduction 1 2 Noncommutative Spaces and Algebras of Functions 5 2.1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Commutative Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Noncommutative Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 The Jacobson (or hull-kernel)Topology . . . . . . . . . . . . . . . . 12 2.4 Compact Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 Noncommutative Lattices 19 3.1 The Topological Approximation . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Order and Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3 How to Recover the Space BeingApproximated . . . . . . . . . . . . . . . 26 3.4 Noncommutative Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.4.1 The space PrimA as a Poset . . . . . . . . . . . . . . . . . . . . . 33 3.4.2 AF-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.4.3 From Bratteli Diagrams to Noncommutative Lattices . . . . . . . . 40 3.4.4 From Noncommutative Lattices to Bratteli Diagrams . . . . . . . . 43 3.5 How to Recover the Algebra BeingApproximated . . . . . . . . . . . . . . 54 3.6 Operator Valued Functions on Noncommutative Lattices . . . . . . . . . . 55 4 Modules as Bundles 57 4.1 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 ProjectiveModules of Finite Type . . . . . . . . . . . . . . . . . . . . . . . 59 4.3 Hermitian Structures over ProjectiveModules . . . . . . . . . . . . . . . . 62 4.4 Few Elements of K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.4.1 The Group K0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.4.2 The K-theory of the Penrose Tiling . . . . . . . . . . . . . . . . . . 68 4.4.3 Higher Order K-groups . . . . . . . . . . . . . . . . . . . . . . . . . 72 5 The Spectral Calculus 75 5.1 In nitesimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75