Derivations on von Neumann algebras are well understood and are always inner, meaning that they act as commutators with a fixed element from the algebra itself. The purpose of this book is to provide a complete description of derivations on algebras of operators affiliated with a von Neumann algebra. The book is designed to serve as an introductory graduate level to various measurable operators affiliated with a von Neumann algebras and their properties. These classes of operators form their respective algebras and the problem of describing derivations on these algebras was raised by Ayupov, and later by Kadison and Liu. A principal aim of the book is to fully resolve the Ayupov-Kadison-Liu problem by proving a necessary and sufficient condition of the existence of non-inner derivation of algebras of measurable operators. It turns out that only for a finite type I von Neumann algebra M may there exist a non-inner derivation on the algebra of operators affiliated with M. In particular, it is established that the classical derivation d/dt of functions of real variables can be extended up to a derivation on the algebra of all measurable functions. This resolves a long-standing problem in classical analysis.

Dr. Aleksey Ber is a researcher with a well-established record (35 publications on MathSciNet, with 191 citations to his work). He is renowned for solving difficult problems in noncommutative analysis and operator theory. Dr. Vladimir Chilin is Emeritus Propessor of Mathematics at the National University of Uzbekistan, Uzbekistan. He is known as one of the most prominent Researchers in Uzbekistan in Functional Analysis (with 158 publications and 605 citations to his work, according to MathSciNet) Dr. Galina Levitina is a Senior Lecturer of Mathematics at the Australian National University, Australia. Galina has 25 papers on MathScinet, has under her belt a paper in Memoirs of European Mathematical Society and 120 citations on MathSciNet) Dr. Fedor Sukochev is Scientia Professor of Mathematics at the University of New South Wales, Australia. He is also a Laureate Professor of Australian Research Council and a member (FAA) of the Australian Academy of Sciences. He has 364 papers and his work cited 3953 times by 755 authors (MathSciNet). Professor Fedor Sukcohev co-authored two books at de Gruyter and fundamental monoraph at Birkhauser (Dodds, Peter G.; de Pagter, Ben; Sukochev, Fedor A.; Noncommutative integration and operator theory. Progress in Mathematics, 349. Birkhäuser/Springer, Cham, [2023]) Dr. Dmitriy Zanin is a Senior Lecturer of Mathematics at the University of New South Wales, Australia. He is extraordinarily productive having 107 publications in top international journals and is ahving 840 citations to his work (MathSciNet). Dr. Zanin co-authored with Professor Sukochev two books published by de Gruyter.