The Domain Algebra of a Cp -semigroup

A CP -semigroup (or quantum dynamical semigroup) is a semigroup φ = {φt : t ≥ 0} of normal completely positive linear maps on B(H), H being a separable Hilbert space, which satisfies φt(1) = 1 for all t and is continuous in the natural sense. Let D be the natural domain of the generator L of φ, φt = exp tL. Since the maps φt 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 : AA ∈ 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 Ω(A) = Ω(A) ⊗A Ω (A), and any linear mapping L : A → B(H) has a symbol σL : Ω (A) → B(H), defined as a linear map by σL(a dx dy) = aL(xy)− axL(y)− aL(x)y + axL(1)y, a, x, y ∈ A. The symbol is a homomorphism of A-bimodules for any ∗-algebra A ⊆ B(H) and any linear map L : A → B(H). When L is the generator of a CP -semigroup with domain algebra A above, we show that the symbol is negative in that σL(ω ω) ≤ 0 for every ω ∈ Ω(A) (−σL is in fact completely positive). Examples are given for which the domain algebraA is, and is not, strongly dense in B(H). We also relate the generator of a CP -semigroup to its commutative paradigm, the Laplacian of a Riemannian manifold. 1. Basic properties of A. Let φ = {φt : t ≥ 0} be a CP -semigroup as defined in the abstract. We first recall four characterizations of the domain of the generator of φ. Lemma 1. Let A ∈ B(H). The following are equivalent. (i) The limit L(A) = lim t→0+ 1 t (φt(A)−A) exists relative to the strong-∗ topology of B(H). On appointment as a Miller Research Professor in the Miller Institute for Basic Research in Science. Support is also acknowledged from NSF grant DMS-9802474 1