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THE DOMAIN ALGEBRA OF A CP-SEMIGROUP William Arveson
 

Summary: THE DOMAIN ALGEBRA OF A CP-SEMIGROUP
William Arveson
Department of Mathematics
University of California
Berkeley CA 94720, USA
18 May, 2000
Abstract. A CP-semigroup (or quantum dynamical semigroup) is a semigroup =
{t : t 0} of normal completely positive linear maps on B(H), H being a separable
Hilbert space, which satisfies t(1) = 1 for all t and is continuous in the natural
sense.
Let D be the natural domain of the generator L of , t = exp tL. Since the maps
t need not be multiplicative D is typically an operator space, but not an algebra.
However, we show that the set of operators
A = {A D : A
A D, AA
D}
is a -subalgebra of B(H), indeed A is the largest self-adjoint algebra contained in
D. Because A is a -algebra one may consider its -bimodule of noncommutative
2-forms 2(A) = 1(A) A 1(A), and any linear mapping L : A B(H) has a
symbol L : 2(A) B(H), defined as a linear map by

  

Source: Arveson, William - Department of Mathematics, University of California at Berkeley

 

Collections: Mathematics