We employ recent advances in the theory of operator spaces, also known as quantized functional analysis, to provide a context in which one can compare categories of modules over operator algebras that are not necessarily self-adjoint. We focus our attention on the category of Hilbert modules over an operator algebra and on the category of operator modules over an operator algebra. The module operations are assumed to be completely bounded - usually, completely contractive. Wedevelop the notion of a Morita context between two operator algebras A and B. This is a system (A,B,{} {A}X {B},{} {B} Y {A},(\cdot,\cdot),[\cdot,\cdot]) consisting of the algebras, two bimodules {A}X {B and {B}Y {A} and pairings (\cdot,\cdot) and [\cdot,\cdot] that induce (complete) isomorphisms betweenthe (balanced) Haagerup tensor products, X \otimes {hB} {} Y and Y \otimes {hA} {} X, and the algebras, A and B, respectively. Thus, formally, a Morita context is the same as that which appears in pure ring theory. The subtleties of the theory lie in the interplay between the pure algebra and the operator space geometry. Our analysis leads to viable notions of projective operator modules and dual operator modules. We show that two C*-algebras are Morita equivalent in our sense if and only ifthey are C*-algebraically strong Morita equivalent, and moreover the equivalence bimodules are the same. The distinctive features of the non-self-adjoint theory are illuminated through a number of examples drawn from complex analysis and the theory of incidence algebras over topological partial orders.Finally, an appendix provides links to the literature that developed since this Memoir was accepted for publication.

During the last few years, the theory of operator algebras, particularly non-self-adjoint operator algebras, has evolved dramatically, experiencing both international growth and interfacing with other important areas. The present volume presents a survey of some of the latest developments in the field in a form that is detailed enough to be accessible to advanced graduate students as well as researchers in the field. Among the topics treated are: operator spaces, Hilbert modules, limit algebras, reflexive algebras and subspaces, relations to basis theory, C* algebraic quantum groups, endomorphisms of operator algebras, conditional expectations and projection maps, and applications, particularly to wavelet theory. The volume also features an historical paper offering a new approach to the Pythagoreans' discovery of irrational numbers.

The study of operator algebras, which grew out of von Neumann's work in the 1920s and the 1930s on modelling quantum mechanics, has in recent years experienced tremendous growth and vitality. This growth has resulted in significant applications in other areas - both within and outside mathematics. The field was a natural candidate for a 1994-1995 program year in Operator Algebras and Applications held at The Fields Institute for Research in the Mathematical Sciences. This volume contains a selection of papers that arose from the seminars and workshops of the program. Topics covered include the classification of amenable C*-algebras, the Baum-Connes conjecture, E[subscript 0] semigroups, subfactors, E-theory, quasicrystals, and the solution to a long-standing problem in operator theory: Can almost commuting self-adjoint matrices be approximated by commuting self-adjoint matrices?

This book is intended for graduate students and research mathematicians interested in calculus of variations and optimal control; optimization.

This title examines in detail graded simple Jordan superalgebras of growth one. Topics include: structure of the even part; Cartan type; even part is direct sum of two loop algebras; $A$ is a loop algebra; and $J$ is a finite dimensional Jordan superalgebra or a Jordan superalgebra of a superform.

This book is intended for graduate students and research mathematicians interested in mechanics of particle systems.

This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.

Let $V = {\mathbb R}^{p,q}$ be the pseudo-Euclidean vector space of signature $(p,q)$, $p\ge 3$ and $W$ a module over the even Clifford algebra $C\! \ell^0 (V)$. A homogeneous quaternionic manifold $(M,Q)$ is constructed for any $\mathfrak{spin}(V)$-equivariant linear map $\Pi : \wedge^2 W \rightarrow V$. If the skew symmetric vector valued bilinear form $\Pi$ is nondegenerate then $(M,Q)$ is endowed with a canonical pseudo-Riemannian metric $g$ such that $(M,Q,g)$ is a homogeneous quaternionic pseudo-Kahler manifold. If the metric $g$ is positive definite, i.e. a Riemannian metric, then the quaternionic Kahler manifold $(M,Q,g)$ is shown to admit a simply transitive solvable group of automorphisms. In this special case ($p=3$) we recover all the known homogeneous quaternionic Kahler manifolds of negative scalar curvature (Alekseevsky spaces) in a unified and direct way. If $p>3$ then $M$ does not admit any transitive action of a solvable Lie group and we obtain new families of quaternionic pseudo-Kahler manifolds. Then it is shown that for $q = 0$ the noncompact quaternionic manifold $(M,Q)$ can be endowed with a Riemannian metric $h$ such that $(M,Q,h)$ is a homogeneous quaternionic Hermitian manifold, which does not admit any transitive solvable group of isometries if $p>3$. The twistor bundle $Z \rightarrow M$ and the canonical ${\mathrm SO}(3)$-principal bundle $S \rightarrow M$ associated to the quaternionic manifold $(M,Q)$ are shown to be homogeneous under the automorphism group of the base. More specifically, the twistor space is a homogeneous complex manifold carrying an invariant holomorphic distribution $\mathcal D$ of complex codimension one, which is a complex contact structure if and only if $\Pi$ is nondegenerate. Moreover, an equivariant open holomorphic immersion $Z \rightarrow \bar{Z}$ into a homogeneous complex manifold $\bar{Z}$ of complex algebraic group is constructed. Finally, the construction is shown to have a natural mirror in the category of supermanifolds. In fact, for any $\mathfrak{spin}(V)$-equivariant linear map $\Pi : \vee^2 W \rightarrow V$ a homogeneous quaternionic supermanifold $(M,Q)$ is constructed and, moreover, a homogeneous quaternionic pseudo-Kahler supermanifold $(M,Q,g)$ if the symmetric vector valued bilinear form $\Pi$ is nondegenerate.

Introduction Calderon weights Applications to real interpolation: reiteration and extrapolation Other classes of weights Extrapolation of weighted norm inequalities via extrapolation theory Applications to function spaces Commutators defined by the K-method Generalized commutators The quasi Banach case Applications to harmonic analysis BMO type spaces associated to Calderon weights Atomic decompositions and duality References.

This book is intended for graduate students and research mathematicians interested in set theory.