Operator Algebras in Dynamical Systems: The Theory of Unbounded Derivations in C*-algebras by Shoichiro Sakai – A Cambri

Introduction

Operator Algebras in Dynamical Systems: The Theory of Unbounded Derivations in C*-algebras by Shoichiro Sakai is a seminal work that bridges advanced mathematics with the foundational needs of quantum physics and statistical mechanics. Published by Cambridge University Press, this hardcover edition is an essential resource for researchers, graduate students, and professionals delving into the intricate world of C*-algebras and their applications. Written by one of the foremost contributors to the field, the book offers a rigorous yet accessible treatment of unbounded derivations, a topic that has profound implications for understanding time evolutions, phase transitions, and the mathematical structure of quantum systems. For Indian readers and students pursuing higher studies in mathematics or theoretical physics, this volume serves as both a comprehensive reference and a gateway to cutting-edge research.

Book Overview

This book systematically explores the theory of unbounded derivations in C*-algebras, a subject that emerged from pressing questions in quantum statistical mechanics. Sakai’s presentation is based on his lectures delivered at Newcastle upon Tyne and Copenhagen, ensuring a pedagogical flow that balances depth with clarity. The text focuses on two major themes: quantum statistical mechanics and differentiations on manifolds. One of its standout achievements is the global construction of derivations for a wide class of interacting models within the C* framework, along with a novel axiomatic approach to building time evolutions and KMS states. The book also aims to formulate the absence theorem of phase transitions in its most general form, making it a cornerstone for theoretical advances in the field.

Key Highlights

• Pioneering Content: Presents the first global construction of derivations for interacting models within C*-algebras.
• Lecture-Based Structure: Derived from actual courses, ensuring a logical and student-friendly progression.
• Focus on Quantum Mechanics: Directly addresses problems in quantum statistical mechanics and phase transitions.
• Rigorous Mathematics: Offers a self-contained treatment suitable for advanced study.
• Authoritative Source: Written by Shoichiro Sakai, a leading figure in operator algebra theory.

Inside the Book

The volume begins with foundational concepts in C*-algebras and derivations, gradually moving to unbounded derivations and their spectral properties. Sakai meticulously develops the theory of KMS states and time evolutions, presenting new axiomatic formulations that simplify complex constructions. Later chapters delve into applications, including the analysis of phase transitions in quantum lattice systems and the differentiation of functions on manifolds. Each chapter includes detailed proofs and examples, making abstract ideas tangible. The book concludes with a discussion of open problems, encouraging readers to engage with ongoing research.

Key Topics

• Unbounded derivations in C*-algebras
• Quantum statistical mechanics and KMS states
• Time evolution and automorphism groups
• Phase transitions and absence theorems
• Derivations on manifolds and differential geometry
• Spectral theory and operator algebras

Reader Benefits

• Deep Understanding: Gain a thorough grasp of unbounded derivations and their role in dynamical systems.
• Research Ready: Equip yourself with tools to tackle advanced problems in operator algebras.
• Practical Insights: Learn how abstract theory applies to concrete physical models.
• Career Advancement: Strengthen your foundation for academic or industrial research in mathematics and physics.

Learning Outcomes

• Master the definition and properties of unbounded derivations in C*-algebras.
• Understand the construction of time evolutions and KMS states in quantum systems.
• Analyze phase transitions using the C* framework.
• Apply derivation theory to manifolds and differential structures.
• Develop the ability to read and contribute to current research literature.

Who Should Read

This book is ideal for graduate students and researchers in mathematics, especially those specializing in operator algebras, functional analysis, or mathematical physics. It is also highly valuable for theoretical physicists working on quantum statistical mechanics, as well as advanced undergraduates with a strong background in analysis. Indian students preparing for competitive exams like GATE, NET, or pursuing PhDs will find it an excellent resource for deepening their expertise.

About the Author

Shoichiro Sakai is a distinguished Japanese mathematician renowned for his pioneering contributions to the theory of C*-algebras and operator algebras. His work has shaped modern mathematical physics, particularly in the study of derivations and quantum dynamical systems. Sakai’s books and lectures are celebrated for their clarity, depth, and originality, making him a revered figure in the global mathematical community.

About the Publisher

Cambridge University Press is one of the world’s oldest and most respected academic publishers, with a history spanning over four centuries. Known for its rigorous editorial standards and commitment to scholarly excellence, Cambridge publishes works that define the frontiers of science, mathematics, and humanities. This hardcover edition reflects their dedication to producing durable, high-quality academic texts.

Conclusion

Operator Algebras in Dynamical Systems: The Theory of Unbounded Derivations in C*-algebras is an indispensable addition to any serious mathematics or physics library. Whether you are a researcher pushing the boundaries of quantum theory or a student building a strong foundation, Sakai’s masterful exposition will guide you through one of the most exciting areas of modern analysis. Order your copy today from Bookshops.in and take a decisive step toward mastering the mathematics of dynamical systems.