This book contains a self-consistent treatment of Besov spaces for W*-dynamical systems, based on the Arveson spectrum and Fourier multipliers. Generalizing classical results by Peller, spaces of Besov operators are then characterized by trace class properties of the associated Hankel operators lying in the W*-crossed product algebra.
These criteria allow to extend index theorems to such operator classes. This in turn is of great relevance for applications in solid-state physics, in particular, Anderson localized topological insulators as well as topological semimetals. The book also contains a self-contained chapter on duality theory for R-actions. It allows to prove a bulk-boundary correspondence for boundaries with irrational angles which implies the existence of flat bands of edge states in graphene-like systems.
This book is intended for advanced students in mathematical physics and researchers alike.
Author(s): Hermann Schulz-Baldes, Tom Stoiber
Series: Mathematical Physics Studies
Publisher: Springer
Year: 2023
Language: English
Pages: 224
City: Cham
Preface and Overview
References
Acronyms and Notations
Contents
1 Preliminaries on Crossed Products
1.1 C*-Dynamical Systems
1.2 W*-Dynamical Systems
1.3 Invariant Traces and GNS Representation
1.4 Arveson Spectrum and Spectral Decomposition
1.5 Dual Traces on Crossed Products
1.6 Dual Action and Duality of Crossed Products
1.7 Spaces of Differentiable Elements
References
2 Besov Spaces for Isometric G-Actions
2.1 Motivation, Definition and Basic Properties of Besov Spaces
2.2 Finite Difference Norm for Besov Spaces
2.3 Differentiability and Besov Spaces
References
3 Quantum Differentiation and Index Theorems
3.1 Besov Spaces and Hankel Operators
3.2 Peller Criterion for Higher Schatten Classes
3.3 Converse of the Peller Criterion
3.4 Breuer Index of Toeplitz Operators
3.5 Sobolev Criterion
References
4 Duality for Toeplitz Extensions
4.1 The Smooth Toeplitz Extension
4.2 Connecting Maps of the Smooth Toeplitz Extension
4.3 Cyclic Cohomology and Smooth Subalgebras
4.4 Duality for Smooth Crossed Products
4.5 Duality of Chern Cocycles
References
5 Applications to Solid State Systems
5.1 Algebraic Set-Up for Solid State Systems
5.2 Half-Space and Boundary Algebras
5.3 Bulk Topological Invariants
5.4 Smooth Bulk-Boundary Correspondence
5.5 Delocalization of Boundary States
5.6 Flat Bands of Edge States
5.7 Application to Graphene
References
Appendix A
A.1 left parenthesis p right parenthesis(p)-Banach-Spaces and Integration
A.2 Non-Commutative upper L Superscript pLp-Spaces
A.3 Complex Interpolation Theory
A.4 Compact and Breuer-Fredholm Operators
Index
