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The Heat Flow of the CCR Algebra
Author(s) -
Arveson William
Publication year - 2002
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609301008396
Subject(s) - mathematics , noncommutative geometry , semigroup , dilation (metric space) , pure mathematics , laplace operator , invariant (physics) , linear map , domain (mathematical analysis) , algebra over a field , discrete mathematics , mathematical analysis , combinatorics , mathematical physics
Let Pf ( x ) = − if ′( x ) and Qf ( x ) = xf ( x ) be the canonical operators acting on an appropriate common dense domain in L 2 (R). The derivations D P ( A ) = i ( PA − AP ) and D Q ( A ) = i ( QA − AQ ) act on the *‐algebra A of all integral operators having smooth kernels of compact support, for example, and one may consider the noncommutative ‘Laplacian’, L =D P 2+D Q 2 , as a linear mapping of A into itself. L generates a semigroup of normal completely positive linear maps on B( L 2 (R)), and this paper establishes some basic properties of this semigroup and its minimal dilation to an E 0 ‐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 ⩽ n < ∞. 2000 Mathematics Subject Classification 46L57 (primary), 46L53, 46L65 (secondary).