Dilations, Completely Positive Maps and Geometry: (84 Texts and Readings in Mathematics)

This book introduces the dilation theory of operators on Hilbert spaces and its relationship to complex geometry. Classical as well as very modern topics are covered in the book. On the one hand, it introduces the reader to the characteristic function, a classical object used by Sz.-Nagy and Foias and still a topic of current research. On the other hand, it describes the dilation theory of the symmetrized bidisc which has been developed mostly in the present century and is a very active topic of research. It also describes an abstract theory of dilation in the setting of set theory. This was developed very recently. A good portion of the book discusses various geometrical objects like the bidisc, the Euclidean unit ball, and the symmetrized bidisc. It shows the similarities and differences between the dilation theory in these domains. While completely positive maps play a big role in the dilation theory of the Euclidean unit ball, this is not so in the symmetrized bidisc for example. There, the central role is played by an operator equation. Targeted to graduate students and researchers, the book introduces the reader to different techniques applicable in different domains.

Table of Contents:
Dilation for One Operator.- C*-Algebras and Completely Positive Maps.- Dilation Theory in Two Variables - The Bidisc.- Dilation Theory in Several Variables - the Euclidean Ball.- The Euclidean Ball - The Drury Arveson Space.- Dilation Theory in Several Variables - The Symmetrized Bidisc.- An Abstract Dilation Theory.

About the Author :
B. V. Rajarama Bhat is Professor at the Theoretical Statistics and Mathematics Division, Indian Statistical Institute, Bengaluru Centre, Karnataka, India. He is Mathematician working in the areas of quantum probability, operator theory, and operator algebras. He is one of the Editors in Chief of the Indian Statistical Institute Series (Springer). He is also Managing Editor of the Infinite Dimensional Analysis, Quantum Probability and Related Topics journal. Tirthankar Bhattacharyya is Professor at the Department of Mathematics, Indian Institute of Science, Bengaluru, Karnataka, India. He is Acclaimed Indian Mathematician who works on the theory of operators in a Hilbert space and its relationship with complex geometry. He is known for his lucid exposition, both in teaching a class and in writing. He serves on the editorial board of the Complex Analysis and Operator Theory journal (Springer) and the Infinite Dimensional Analysis, Quantum Probability and Related Topics journal.

Review :
“In this book, the authors obtain concrete constructions of dilations of commuting operator tuples related to four domains where this theory has been beautifully successful so far. … The constructions in this book are very interesting because of various inputs that are required, like, for example, the Fejér-Riesz theorem. So, it connects to holomorphic function theory and geometry very well.” (Luo Yi Shi, Mathematical Reviews, February, 2025)