2000/05/24 by William Arveson, Arveson, William
Computer Science · Decision Sciences · Mathematics · #Advanced Bandit Algorithms Research #FOS: Mathematics #Operator Algebras (math.OA) #Polynomial and algebraic computation #Quantum Computing Algorithms and Architecture #math.OA
paper · pdf · doi:10.48550/arxiv.math/0005251
19 pages typeset
arxiv created 2000/05/24 · openalex publication_date 2000/05/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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).