C*-Algebras and W*-Algebras (Classics in Mathematics) 1998th Edition by Shoichiro Sakai (PDF)

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From the reviews: “This book is an excellent and comprehensive survey of the theory of von Neumann algebras. It includes all the fundamental results of the subject, and is a valuable reference for both the beginner and the expert.” Mathematical Reviews

User’s Reviews

Editorial Reviews: Review From the reviews: “This book is an excellent and comprehensive survey of the theory of von Neumann algebras. It includes all the fundamental results of the subject, and is a valuable reference for both the beginner and the expert.” (Math. Reviews) “In theory, this book can be read by a well-trained third-year graduate student – but the reader had better have a great deal of mathematical sophistication. The specialist in this and allied areas will find the wealth of recent results and new approaches throughout the text especially rewarding.” (American Scientist) “The title of this book at once suggests comparison with the two volumes of Dixmier and the fact that one can seriously make this comparison indicates that it is a far more substantial work than others on this subject which have recently appeared”(BLMSoc) About the Author Biography of Shôichirô SakaiShôichirô Sakai was born in 1928 in Kanuma, Japan. He received his B. A. in mathematics in 1953 and his doctorate in 1961 from Tohoku University, Sendai. He was a research assistant there (195-1960), then a faculty member of Waseda University, Tokyo (1960-1964). After 2 years on the faculty of the University of Pennsylvania, he became a professor in 1966 and stayed there till 1979 when he returned to Japan to a chair at Nihon University, Tokyo. Prof. Sakai has also held visiting positions at Yale University (1962-1964) and at MIT (1967-1968). His main interests are in operator algebras, functional analysis and mathematical physics.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This book is excellent. One of the fundamental works of the 20th century, which will likely be of great value throughout this century.

⭐This is a classical monograph on desk of each mathematician and graduate student at using operator algebraic, especially von Neumann algebras

⭐Theorem, proof, theorem, proof, theorem, proof, … qed.One has to admire the scope of this book. It really does cover nearly all major results in the subject up until that time. However, in order to accomplish this goal without expanding into multiple volumes, the author omits examples, motivation, and entire lemmas worth of ideas used in proofs. J.E. Littlewood once said that “two trivialities omitted can add up to an impasse”. He would shutter at some of leaps of reasoning required of the reader. Theorems are rarely cited and ideas are lifted from the middle of proofs and utilized several sections later without reference. Recommended for the masochist mathematician with interest in operator theory.

⭐Together with Dixmier’s two volumes, I used this text extensively while writing my Ph.D dissertation onideal theory in C*-Algebras.I find derivation andautomorphims of C*- and W*-Algebras and a Stone-Weierstrass Theorem for C*-Algebras of great interest. Since this publication, *-Algebras have become ubiquitous in mathematical physics. Nevertheless, I return to Sakai (and Dixmier) for the great foundational work in the subject. Prerequisites include solid understanding of functional analysis at the graduate level.

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