An Introduction to Noncommutative Spaces and Their Geometries

1. Front Matter

   Pages I-XV

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2. Introduction

   Pages 1-5

3. Noncommutative Spaces and Algebras of Functions

   Pages 7-20

4. Projective Systems of Noncommutative Lattices

   Pages 21-58

5. Modules as Bundles

   Pages 59-68

6. A Few Elements of K-Theory

   Pages 69-81

7. The Spectral Calculus

   Pages 83-103

8. Noncommutative Differential Forms

   Pages 105-121

9. Connections on Modules

   Pages 123-132

10. Field Theories on Modules

    Pages 133-150

11. Gravity Models

    Pages 151-163

12. Quantum Mechanical Models on Noncommutative Lattices

    Pages 165-169

13. Back Matter

    Pages 171-207

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These lecture notes are an introduction to several ideas and applications of noncommutative geometry. It starts with a not necessarily commutative but associative algebra which is thought of as the algebra of functions on some 'virtual noncommutative space'. Attention is switched from spaces, which in general do not even exist, to algebras of functions. In these notes, particular emphasis is put on seeing noncommutative spaces as concrete spaces, namely as a collection of points with a topology. The necessary mathematical tools are presented in a systematic and accessible way and include among other things, C'*-algebras, module theory and K-theory, spectral calculus, forms and connection theory. Application to Yang--Mills, fermionic, and gravity models are described. Also the spectral action and the related invariance under automorphism of the algebra is illustrated. Some recent work on noncommutative lattices is presented. These lattices arose as topologically nontrivial approximations to 'contuinuum' topological spaces. They have been used to construct quantum-mechanical and field-theory models, alternative models to lattice gauge theory, with nontrivial topological content. This book will be essential to physicists and mathematicians with an interest in noncommutative geometry and its uses in physics.

• algebra
• algebras
• field theories
• geometry
• gravity
• gravity models
• noncommunative lattices
• noncommutative spaces
• quantum mechanical models
• spectra
• topology

• Dipartimento di Scienze Matematiche, Università degli Studi di Trieste, Trieste, Italy

  Giovanni Landi

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