Геометрия квантового расстояния Громова--Хаусдорфа. ч.1

Metric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory. The new wave began with seminal papers by Svarc and Milnor on the growth of groups and the spectacular proof of the rigidity of lattices by Mostow. This progress was followed by the creation of the asymptotic metric theory of infinite groups by Gromov. The structural metric approach to the Riemannian category, tracing back to Cheeger's thesis, pivots around the notion of the Gromov–Hausdorff distance between Riemannian manifolds. This distance organizes Riemannian manifolds of all possible topological types into a single connected moduli space, where convergence allows the collapse of dimension with unexpectedly rich geometry, as revealed in the work of Cheeger, Fukaya, Gromov and Perelman. Also, Gromov found metric structure within homotopy theory and thus introduced new invariants controlling combinatorial complexity of maps and spaces, such as the simplicial volume, which is responsible for degrees of maps between manifolds. During the same period, Banach spaces and probability theory underwent a geometric metamorphosis, stimulated by the Levy–Milman concentration phenomenon, encompassing the law of large numbers for metric spaces with measures and dimensions going to infinity. The first stages of the new developments were presented in Gromov's course in Paris, which turned into the famous'Green Book'by Lafontaine and Pansu (1979). The present English translation of that work has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices – by Gromov on Levy's inequality, by Pansu on'quasiconvex'domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures – as well as an extensive bibliographyand index round out this unique and beautiful book. Предварительные результаты Банаховы пространства Направленности и их пределы Слабая топология *-слабая топология Рефлексивные пространства Сепарабельные пространства Равномерная выпуклость Гильбертовы пространства Банаховы алгебры Элементы теории алгебр Алгебра, унитальная алгебра Нормированные и унитальные нормированные алгебры Банаховы алгебры Идеалы, модулярные идеалы Гомоморфизмы алгебр Унитализация Резольвентные множества и спектры в унитальной алгебре Случай унитальных банаховых алгебр Спектральный радиус в унитальной алгебре Спектры и спектральные радиусы в неунитальной алгебре Экспоненты в унитальной банаховой алгебре Модулярные идеалы, продолжение Характеры коммутативной алгебры, ее спектр Характеры коммутативной алгебры и ее унитализации Характеры, спектры, топология пространства характеров Отождествления Представление Гельфанда коммутативной банаховой алгебры Элементы теории C*-алгебр Алгебры с инволюцией или *-алгебры Нормированные и банаховы *-алгебры C*-алгебры Унитализация банаховой *-алгебры и C*-алгебры Представление Гельфанда коммутативной C*-алгебры Некоторые приложения представления Гельфанда Литература This book is an English translation of the famous "Green Book" by Lafontaine and Pansu (1979). It has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices, by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures, as well as an extensive bibliography and index round out this unique and beautiful book. This book explores exciting new connections between geometry and probability theory, as well as their links to analysis. This well-written book includes numerous illustrations and examples and will serve as a valuable resource for geometers, analysts, and probabilists. In classical Riemannian geometry, one begins with a C manifold X and then studies smooth, positive-definite sections g of the bundle S2T X.