Categories Of Operator Modules Morita Equivalence And Projective Modules
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Categories of Operator Modules (Morita Equivalence and Projective Modules)
Author: David P. Blecher
language: en
Publisher: American Mathematical Soc.
Release Date: 2000
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.
Categories of Operator Modules (Morita Equivalence and Projective Modules)
The authors employ 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. Attention is focused 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.
Operator Algebras and Their Applications
Author: Peter A. Fillmore
language: en
Publisher: American Mathematical Soc.
Release Date: 1997
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 programme 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 programme. Topics covered include the classification of amenable C ]*-algebras, the Baum-Connes conjecture, E [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?