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Operator Algebras and Their Modules: An operator space approach

David P. Blecher and Christian Le Merdy

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198526599
eISBN:
9780191712159
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526599.001.0001
Subject:
Mathematics, Pure Mathematics

This book presents the general theory of algebras of operators on a Hilbert space, and the modules over such algebras. The new theory of operator spaces is presented early on and the text assembles ... More


Basic theory of operator algebras

David P. Blecher and Christian Le Merdy

in Operator Algebras and Their Modules: An operator space approach

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198526599
eISBN:
9780191712159
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526599.003.0002
Subject:
Mathematics, Pure Mathematics

This chapter begins with a discussion of operator algebras and unitizations. It then covers some basic constructions, the abstract characterization of operator algebras, universal constructions of ... More


Interpolation and Definability: Modal and Intuitionistic Logics

Dov M. Gabbay and Larisa Maksimova

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198511748
eISBN:
9780191705779
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198511748.001.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This book focuses on interpolation and definability. This notion is not only central in pure logic, but has significant meaning and applicability in all areas where logic itself is applied, ... More


Set Theory: Boolean-Valued Models and Independence Proofs

John L. Bell

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198568520
eISBN:
9780191717581
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568520.001.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This is the third edition of a well-known graduate textbook on Boolean-valued models of set theory. The aim of the first and second editions was to provide a systematic and adequately motivated ... More


The Greate Invention of Algebra: Thomas Harriot's Treatise on equations

Jacqueline A. Stedall

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198526025
eISBN:
9780191712364
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526025.001.0001
Subject:
Mathematics, History of Mathematics

This book casts new light on the work of Thomas Harriot (c.1560-1621), an innovative thinker and practitioner in several branches of the mathematical sciences, including navigation, astronomy, ... More


Some ‘extremal theory’

David P. Blecher and Christian Le Merdy

in Operator Algebras and Their Modules: An operator space approach

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198526599
eISBN:
9780191712159
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526599.003.0004
Subject:
Mathematics, Pure Mathematics

Many problems in functional analysis are best addressed via extreme points. Extreme points, particularly in the guise of the Choquet or Shilov boundaries, play a substantial role in the theory of ... More


Completely isomorphic theory of operator algebras

David P. Blecher and Christian Le Merdy

in Operator Algebras and Their Modules: An operator space approach

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198526599
eISBN:
9780191712159
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526599.003.0005
Subject:
Mathematics, Pure Mathematics

This chapter studies operator algebras ‘up to isomorphism’ or ‘up to complete isomorphism’. Topics covered include homomorphisms of operator algebras, completely bounded characterizations, examples ... More


Basic theory of operator modules

David P. Blecher and Christian Le Merdy

in Operator Algebras and Their Modules: An operator space approach

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198526599
eISBN:
9780191712159
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526599.003.0003
Subject:
Mathematics, Pure Mathematics

This chapter discusses the basics of operator modules. Topics covered include Hilbert modules, operator modules over operator algebras, two module tensor products, module maps, module map extension ... More


Tensor products of operator algebras

David P. Blecher and Christian Le Merdy

in Operator Algebras and Their Modules: An operator space approach

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198526599
eISBN:
9780191712159
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526599.003.0006
Subject:
Mathematics, Pure Mathematics

Tensor products and C*-norms play a prominent role in the theory of C*-algebras, in particular in the study of nuclear C*-algebras and semidiscrete (or injective) von Neumann algebras. This chapter ... More


Selfadjointness criteria

David P. Blecher and Christian Le Merdy

in Operator Algebras and Their Modules: An operator space approach

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198526599
eISBN:
9780191712159
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526599.003.0007
Subject:
Mathematics, Pure Mathematics

This chapter discusses several interesting criteria which force an operator algebra to be selfadjoint. Such results are in some sense ‘negative’ in nature, showing that certain results or themes ... More


Visual Thinking in Mathematics

Marcus Giaquinto

Published in print:
2007
Published Online:
September 2007
ISBN:
9780199285945
eISBN:
9780191713811
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199285945.001.0001
Subject:
Philosophy, Logic/Philosophy of Mathematics

Visual thinking — visual imagination or perception of diagrams and symbol arrays, and mental operations on them — is omnipresent in mathematics. Is this visual thinking merely a psychological aid, ... More


The mathematics of John Pell

Jacqueline Stedall

in John Pell (1611-1685) and His Correspondence with Sir Charles Cavendish: The Mental World of an Early Modern Mathematician

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198564843
eISBN:
9780191713750
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198564843.003.0009
Subject:
Mathematics, History of Mathematics

This second part of the book presents the mathematical works of John Pell. Pell's published works were few: the Controversiae pars prima in 1647, An Introduction to Algebra in 1668, and the Table of ... More


BOOLEAN AND HEYTING ALGEBRAS: THE ESSENTIALS

John L. Bell

in Set Theory: Boolean-Valued Models and Independence Proofs

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198568520
eISBN:
9780191717581
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568520.003.0001
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter provides a brief account of the theory of Boolean and Heyting algebras, including the basic representation theorems and their connections with logic.


MONADS AND ALGEBRAS

Steve Awodey

in Category Theory

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198568612
eISBN:
9780191717567
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198568612.003.0010
Subject:
Mathematics, Algebra

This chapter presents a third characterization of adjunctions. This one has the virtue of being entirely equational. Topics discussed include the triangle identities, monads and adjoints, algebras ... More


MODAL AND SUPERINTUITIONISTIC LOGICS: BASIC CONCEPTS

D.M. Gabbay and L. Maksimova

in Interpolation and Definability: Modal and Intuitionistic Logics

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198511748
eISBN:
9780191705779
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198511748.003.0002
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter presents basic definitions and known results related to modal and superintuitionistic logics. Well-known Hilbert-style axiomatizations of the most known modal systems K, D, T, K4, D4, ... More


How algebra was entertained and cultivated in Europe

Jacqueline A. Stedall

in A Discourse Concerning Algebra: English Algebra to 1685

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198524953
eISBN:
9780191711886
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198524953.003.0002
Subject:
Mathematics, History of Mathematics

This chapter discusses the development of algebra in Europe. Topics covered include the origins from which European algebra began to evolve and from which it took its name; The Liber abbaci (1202) of ... More


AN INTRODUCTION TO THE THEORY OF C*-ALGEBRAS IN CONSTRUCTIVE MATHEMATICS

Hiroki Takamura

in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198566519
eISBN:
9780191713927
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198566519.003.0018
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter introduces an elementary theory of C*-algebras in the context of Bishop-style constructive mathematics. It givens proof of the Gelfand-Naĭmark-Segal (GNS) construction theorem in ... More


3-fold flips after Shokurov

Alessio Corti

in Flips for 3-folds and 4-folds

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198570615
eISBN:
9780191717703
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198570615.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter gives a concise, complete, and pedagogical proof of existence of 3-fold flips according to Shokurov. In particular, the foundation of the theory of b-divisors, algebras of rational ... More


What Does It Take to Be Human?: Lessons From High School Algebra

John R. Anderson

in How Can the Human Mind Occur in the Physical Universe?

Published in print:
2007
Published Online:
September 2007
ISBN:
9780195324259
eISBN:
9780199786671
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195324259.003.0005
Subject:
Psychology, Cognitive Models and Architectures

Much of our cognitive architecture we share with other primates. The unique feature of the human mind is the ability to acquire nearly arbitrary competences that were not anticipated in our ... More


Understanding the Properties of “Numbers”

Victor J. Katz and Karen Hunger Parshall

in Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691149059
eISBN:
9781400850525
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691149059.003.0013
Subject:
Mathematics, History of Mathematics

This chapter looks at how mathematicians sought to understand the properties of “numbers” and in doing so pave the way for modern algebra. As mathematicians in the eighteenth and nineteenth centuries ... More


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