The Characterization of Differential Operators by Locality: Dissipations and Ellipticity

Abstract

Let δ be the generator of a C₀-group of *-automorphisms of a C^*-algebra \mathcal{A} and H a differential operator of the form
H=∑\limits_m=0^p λ_mδ^m,
where λ_m∈C. It is known from a previous work that if \mathcal{A} is abelian then H is a dissipation, i. e.
H(a^*a){≤}a^*H(a)+H(a^*)a, a∈D(H),
if, and only if, λ_m=0 for m>2, λ₂{≤}0, and λ₀{≥}0. This conclusion is no longer generally true for non-abelian \mathcal{A}, but it is true in a variety of special cases which we discuss, e. g. if \mathcal{A} is isomorphic to the C^*-algebra of all compact operators on a Hilbert space \mathcal{H} and σ is unbounded.