Jump to ContentJump to Main Navigation
Quantum Dynamical Systems$
Users without a subscription are not able to see the full content.

Robert Alicki and Mark Fannes

Print publication date: 2001

Print ISBN-13: 9780198504009

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780198504009.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2020. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 28 May 2020

Algebraic Tools

Algebraic Tools

Chapter:
(p.82) 5 Algebraic Tools
Source:
Quantum Dynamical Systems
Author(s):

Robert Alicki

Mark Fannes

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198504009.003.0005

This chapter introduces an abstract algebraic language unifying the description of classical systems, finite quantum systems, and infinite ones, the last appearing in particle and statistical physics. In the first part, the general theory of C*-algebras is presented and illustrated by the following examples: general finite dimensional algebras, Abelian algebras and Gelfand's theorem, UHF-algebras, and algebras generated by group representations such as the CCR-algebra arising from the Heisenberg group. Then the theory of states on C*-algebras leading to the GNS-representation in terms of operators on Hilbert spaces is outlined. The basic notion of algebraic dynamical system is given in terms of automorphisms on a C*-algebra of observables and the link to the Hilbert space formalism based on unitary operators is provided by the theory of von Neumann algebras. The examples of the Koopman formalism and the rotation algebra are worked out.

Keywords:   C*-algebra, Gelfand's theorem, uhf-algebra, group algebra, Heisenberg group, CCR-algebra, GNS-representation theorem, algebraic dynamical system, von Neumann algebra, Koopman formalism

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .