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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
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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

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