The domain algebra of a CP semigroup

Abstract

A CP semigroup is a semigroup of normal unit-preserving completely positive maps acting on the algebra B(H) of all operators on a separable Hilbert space H. Such a semigroup has a natural generator L; since the individual maps of the semigroup need not be multiplicative, the domain Dom(L) of L is typically an operator system, but not an algebra. However, we show that the set of all operators A in Dom(L), with the property that both A*A and AA* belong to Dom(L), is a *-algebra, called the domain algebra of the CP semigroup. Using this algebra, it is possible to draw a very close parallel with the Laplacian of a Riemannian manifold. We discuss properties of the "symbol" of L as the noncommutative counterpart of a (semidefinite) Riemannian metric, and give examples for which the domain algebra is, and is not, strongly dense in B(H).