Relative locality of derivations

Abstract

Let H and K be symmetric linear operators on a C*-algebra U with domains D(H) and D(K). H is defined to be strongly K-local if ω(K(A)*K(A)) = 0 implies ω(H(A)* H(A)) = 0 for A ε{lunate} D(H) ∩ D(K) and ω in the state space of U, and H is completely strongly K-local if Ω(K(A)*K(A))=0 implies Ω(H(A)*H(A))=0 for A ∈ D(H) ∩ D(K) and Ω in the state of U, and H is cpmpletely strongly K-local if H⊗{up harpoon left}n is K⊗{up harpoon left}n-local on U⊗Mn for all n ≥ 1, where 1n is the identity on the n × n matrices Mn. If U is abelian then strong locality and complete strong locality are equivalent. The main result states that if τ is a strongly continuous one-parameter group of *-automorphisms of U with generator δ0 and δ is a derivation which commutes with τ and is completely strongly δ0-local then δ generates a group α of *-automorphisms of U. Various characterizations of α are given and the particular case of periodic τ is discussed. Show more