## 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

## Operator spaces

*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/9780198526599.003.0001
- Subject:
- Mathematics, Pure Mathematics

This chapter presents some background results about operator spaces and establishes some notations which will be used throughout this book. It can serve as a mini-course on the basics of operator ... 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

## Basic Tools For Quantum Mechanics

*Robert Alicki and Mark Fannes*

### in Quantum Dynamical Systems

- Published in print:
- 2001
- Published Online:
- February 2010
- ISBN:
- 9780198504009
- eISBN:
- 9780191708503
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504009.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave ... More

## Quantum Quandaries: A Category-Theoretic Perspective

*John Baez*

### in The Structural Foundations of Quantum Gravity

- Published in print:
- 2006
- Published Online:
- October 2011
- ISBN:
- 9780199269693
- eISBN:
- 9780191699436
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199269693.003.0008
- Subject:
- Philosophy, Philosophy of Science

Part of the difficulty of combining general relativity and quantum theory is that they use different sorts of mathematics: one is based on objects such as manifolds, the other on objects such as ... 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

## THE PME AS AN ABSTRACT EVOLUTION EQUATION. SEMIGROUP APPROACH

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0010
- Subject:
- Mathematics, Mathematical Physics

This chapter addresses the problem of the construction of solutions of the GPME by viewing it as an abstract evolution equation, or more precisely as an ordinary differential equation with values in ... More

## Differentiability of Lipschitz Maps on Hilbert Spaces

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0016
- Subject:
- Mathematics, Analysis

This chapter presents a separate, essentially self-contained, nonvariational proof of existence of points of Fréchet differentiability of R²-valued Lipschitz maps on Hilbert spaces. It begins with ... More

## Quantizing

*Laura Ruetsche*

### in Interpreting Quantum Theories

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199535408
- eISBN:
- 9780191728525
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535408.003.0002
- Subject:
- Philosophy, Philosophy of Science

This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter ... More

## Stochastic Dynamics In Hilbert Space

*Heinz-Peter Breuer and Francesco Petruccione*

### in The Theory of Open Quantum Systems

- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.06
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter contains the foundations of the stochastic formulation of continuous measurements carried out on an open quantum system. It shows that the time evolution of the state vector, conditioned ... More

## Probability Distributions on Hilbert Space

*Heinz-Peter Breuer and Francesco Petruccione*

### in The Theory of Open Quantum Systems

- Published in print:
- 2007
- Published Online:
- February 2010
- ISBN:
- 9780199213900
- eISBN:
- 9780191706349
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199213900.003.05
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter characterizes the statistical properties of a quantum mechanical ensemble in terms of a density matrix. However, if selective measurements of one or several observables are carried out ... More

## The Basic Theory of Quantum Mechanics

*Bas C. van Fraassen*

### in Quantum Mechanics: An Empiricist View

- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198239802
- eISBN:
- 9780191597466
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198239807.003.0006
- Subject:
- Philosophy, Philosophy of Science

Covered are Hilbert space, vector, and operator representations of pure and mixed states, measurable physical quantities (observables), Gleason's theorem, Lueders’ Rule, unitary operators, and ... More

## Fr ´Echet Differentiability of Vector-Valued Functions

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0013
- Subject:
- Mathematics, Analysis

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus o(tⁿ logⁿ⁻¹(1/t)) then every Lipschitz map of X to a space of dimension not exceeding n has ... More

## Asymptotic Fr echet ´Differentiability

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0015
- Subject:
- Mathematics, Analysis

This chapter presents the current development of the first, unpublished proof of existence of points Fréchet differentiability of Lipschitz mappings to two-dimensional spaces. For functions into ... More

## Monoidal 2-Categories

*Chris Heunen and Jamie Vicary*

### in Categories for Quantum Theory: An Introduction

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780198739623
- eISBN:
- 9780191802584
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198739623.003.0008
- Subject:
- Mathematics, Mathematical Physics, Applied Mathematics

Monoidal 2-categories are higher-dimensional versions of monoidal categories, allowing a more expressive syntax that plays an important role in modern mathematics. We explore their two-dimensional ... More

## Abstract Hilbert Space

*John von Neumann*

*Nicholas A. Wheeler (ed.)*

### in Mathematical Foundations of Quantum Mechanics: New Edition

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780691178561
- eISBN:
- 9781400889921
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691178561.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter defines Hilbert space, which furnishes the mathematical basis for the treatment of quantum mechanics. This is done within the context of an equation introduced in the previous chapter, ... More

## Quantum Many-Particle Systems and Second Quantization

*Hans-Peter Eckle*

### in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

Chapter 2 provides a review of pertinent aspects of the quantum mechanics of systems composed of many particles. It focuses on the foundations of quantum many-particle physics, the many-particle ... More

## Probability in Hilbert Space

*Jochen Rau*

### in Quantum Theory: An Information Processing Approach

- Published in print:
- 2021
- Published Online:
- July 2021
- ISBN:
- 9780192896308
- eISBN:
- 9780191918865
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192896308.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

This chapter introduces the mathematical framework, basic rules, and some key results of quantum theory. After a succinct overview of linear algebra and an introduction to complex Hilbert space, it ... More

## Basics

*Chris Heunen and Jamie Vicary*

### in Categories for Quantum Theory: An Introduction

- Published in print:
- 2019
- Published Online:
- January 2020
- ISBN:
- 9780198739623
- eISBN:
- 9780191802584
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198739623.003.0009
- Subject:
- Mathematics, Mathematical Physics, Applied Mathematics

This book assumes familiarity with some basic ideas from category theory, linear algebra and quantum computing. This self-contained chapter gives a quick summary of the essential aspects of these ... More

## Hilbert Spaces

*Adel N. Boules*

### in Fundamentals of Mathematical Analysis

- Published in print:
- 2021
- Published Online:
- May 2021
- ISBN:
- 9780198868781
- eISBN:
- 9780191905254
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198868781.003.0007
- Subject:
- Mathematics, Analysis, Applied Mathematics

This chapter is a good introduction to Hilbert spaces and the elements of operator theory. The two leading sections contains staple topics such as the projection theorem, projection operators, the ... More