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An Introduction to Noncommutative Spaces and Their Geometries (Paperback, 1997 ed.)
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An Introduction to Noncommutative Spaces and Their Geometries (Paperback, 1997 ed.)
Series: Lecture Notes in Physics Monographs, 51
Expected to ship within 10 - 15 working days
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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.
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