In algebraic topology some classical invariants - such as Betti numbers and Reidemeister torsion - are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to non-compact spaces and infinite groups. These new L 2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K -Theory, differential geometry, non-commutative geometry and spectral theory. It is particularly these interactions with different fields that make L 2-invariants very powerful and exciting. The book presents a comprehensive introduction to this area of research, as well as its most recent results and developments. It is written in a way which enables the reader to pick out a favourite topic and to find the result she or he is interested in quickly and without being forced to go through other material.

A comprehensive introduction to the field of L2-Invariants Presents the most recent results and developments Chapters are kept as independent of one other as possible Each chapter includes exercises; hints to their solution are given Contains an extensive index The book will become a standard reference work in the field of L2-invariants Includes supplementary material: sn.pub/extras

Autorentext

Wolfgang Lück has worked on topology, K-theory, and global analysis. He completed his PhD in 1984 under the supervision of Prof. Tammo tom Dieck at Göttingen, where he also obtained the venia legendi in 1989. He has held permanent positions at the universities at Lexington, Mainz and Münster, and is currently Professor at the University of Bonn. He was awarded the Max Planck Research Award in 2003, the Gottfried Wilhelm Leibniz Award in 2008 and the von Staudt Prize in 2025. His other honors include membership of the Leopoldina (since 2010) and of the Nordrhein-Westfälische Akademie der Wissenschaft und der Künste (since 2013), Fellowship of the American Mathematical Society (since 2013), and he was president of the Deutsche Mathematiker Vereinigung during 2009–2010. In addition, he was a Max Planck Fellow from 2013–2023 and obtained an ERC Advanced Grant in 2014. To date, he has directed the theses of 30 PhD students. In Bonn, he was the director of the Hausdorff Institute from 2011–2017 and the spokesperson of the Cluster of Excellence Hausdorff Center for Mathematics from 2019–2022. He has been married to Sibylle Lück since 1984 and has four children and four grandchildren.

Inhalt

0. Introduction.- 1. L2-Betti Numbers.- 2. Novikov-Shubin Invariants.- 3. L2-Torsion.- 4. L2-Invariants of 3-Manifolds.- 5. L2-Invariants of Symmetric Spaces.- 6. L2-Invariants for General Spaces with Group Action.- 7. Applications to Groups.- 8. The Algebra of Affiliated Operators.- 9. Middle Algebraic K-Theory and L-Theory of von Neumann Algebras.- 10. The Atiyah Conjecture.- 11. The Singer Conjecture.- 12. The Zero-in-the-Spectrum Conjecture.- 13. The Approximation Conjecture and the Determinant Conjecture.- 14. L2-Invariants and the Simplicial Volume.- 15. Survey on Other Topics Related to L2-Invariants.- 16. Solutions of the Exercises.- References.- Notation.