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Noncommutative geometry is a novel approach which is opening up possibilities for geometry from a mathematical viewpoint. Developments in string theory have supported the idea of quantum spaces, and have strongly stimulated the research in this field. This volume contains survey lectures and research articles which address these issues and topics.

This text provides an overview of operator algebra theory and an introduction to basic tools used in noncommunicative geometry. The book concludes with applications of operator algebras to Atiyah-Singer type index theorems.