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This book provides a comprehensive introduction to the systematic theory of tensor products and tensor norms within the framework of operator spaces. The use of tensor products has significantly advanced functional analysis and other areas of mathematics and physics, and the field of operator spaces is no exception. Building on the theory of tensor products in Banach spaces, this work adapts the definitions and results to the operator space context. This approach goes beyond a mere translation of existing results. It introduces new insights, techniques, and hypotheses to address the many challenges of the non-commutative setting, revealing several notable differences to the classical theory. This text is expected to be a valuable resource for researchers and advanced students in functional analysis, operator theory, and related fields, offering new perspectives for both the mathematics and physics communities. By presenting several open problems, it also serves as a potential source for further research, particularly for those working in operator spaces or operator algebras.
List of contents
Chapter 1. Preliminaries.- Chapter 2. Introduction to operator space tensor norms.- Chapter 3. Finite and cofinite hulls.- Chapter 4. The five basic lemmas.- Chapter 5. Dual operator space tensor norms.- Chapter 6. The completely bounded approximation property.- Chapter 7. Mapping ideals.- Chapter 8. Maximal operator space mapping ideals.- Chapter 9. Minimal operator space mapping ideals.- Chapter 10. Completely projective/injective tensor norms.- Chapter 11. Injective/projective hulls and accessibility.- Chapter 12. Natural operator space tensor norms.- Chapter 13. Conclusions and some open questions.
About the author
Javier Alejandro Chávez-Domínguez
is an Associate Professor in the Department of Mathematics of the University of Oklahoma (USA). His main research interest is Functional Analysis with an emphasis on its non-linear and non-commutative aspects, particularly Operator Spaces, Tensor Products and Operator Ideals, and Quantum Graphs/Metric Spaces.
Verónica Dimant
is a Full Professor at the University of San Andrés (Argentina) and Independent Researcher at CONICET. Her research interest lies in Non-linear Functional Analysis with a focus on Holomorphy, Polynomials and Tensor Products in Banach spaces and operator spaces.
Daniel Galicer
is an Associate Professor at Universidad Torcuato Di Tella (Argentina) and an Independent Researcher at IMAS-CONICET. His research focuses on Functional Analysis, with an emphasis on the Local Theory of Banach Spaces, Infinite-Dimensional Analysis, Asymptotic Geometric Analysis, Tensor Products, and the interactions between Analysis and Probability.
Summary
This book provides a comprehensive introduction to the systematic theory of tensor products and tensor norms within the framework of operator spaces. The use of tensor products has significantly advanced functional analysis and other areas of mathematics and physics, and the field of operator spaces is no exception. Building on the theory of tensor products in Banach spaces, this work adapts the definitions and results to the operator space context. This approach goes beyond a mere translation of existing results. It introduces new insights, techniques, and hypotheses to address the many challenges of the non-commutative setting, revealing several notable differences to the classical theory. This text is expected to be a valuable resource for researchers and advanced students in functional analysis, operator theory, and related fields, offering new perspectives for both the mathematics and physics communities. By presenting several open problems, it also serves as a potential source for further research, particularly for those working in operator spaces or operator algebras.
