Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-30T19:55:41.108Z Has data issue: false hasContentIssue false
  • English
  • Français

THE HEAT FLOW OF THE CCR ALGEBRA

Published online by Cambridge University Press:  18 January 2002

WILLIAM ARVESON
WILLIAM ARVESON
Affiliation:
Department of Mathematics, University of California, Berkeley CA 94720, U.S.A.; arveson@math.berkeley.edu
Get access

Abstract

Let Pf(x) =−if′(x) and Qf(x) = xf(x) be the canonical operators acting on an appropriate common dense domain in L2(ℝ). The derivations DP(A) = i(PA−AP) and DQ(A) = i(QA−AQ) act on the *-algebra [Ascr ] of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative ‘Laplacian’, L = D2P+D2Q, as a linear mapping of [Ascr ] into itself.

L generates a semigroup of normal completely positive linear maps on [Bscr ](L2(ℝ)), and this paper establishes some basic properties of this semigroup and its minimal dilation to an E0-semigroup. In particular, the author shows that its minimal dilation is pure and has no normal invariant states, and he discusses the significance of those facts for the interaction theory introduced in a previous paper.

There are similar results for the canonical commutation relations with n degrees of freedom, where 1 [les ] n < 1.

Type
NOTES AND PAPERS
Information
Bulletin of the London Mathematical Society , Volume 34 , Issue 1 , January 2002 , pp. 73 - 83
Copyright
© The London Mathematical Society 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)