 
Summary: THE DOMAIN ALGEBRA OF A CPSEMIGROUP
William Arveson
Department of Mathematics
University of California
Berkeley CA 94720, USA
18 May, 2000
Abstract. A CPsemigroup (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 selfadjoint algebra contained in
D. Because A is a algebra one may consider its bimodule of noncommutative
2forms 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
