Extensions of unbounded ∗-derivations in UHF C∗-algebras

AbstractWe consider unbounded ∗-derivations δ in UHF-C∗-algebras A=(∪∞n=1An)−) with dense domain. If ϕn:A→An denotes the conditional expectations onto the finite type I factors An, then we introduce a weak-commutativity condition for δ and the sequence (ϑn). As a consequence of this condition on δ we establish the existence of an extension derivation δ′ which is the infinitesimal generator of a strongly continuous one-parameter group, α: R → Aut(A), of ∗-automorphisms, i.e., δ′(x) = (ddt)αt(x)¦t = 0 for x ϵ D(δ′). Special properties of α (alias δ′) are considered. We show that AF-algebras are associated to proper restrictions δ of derivations δ′ of product type. We then turn to the extendability problem for quasifree derivations in the CAR-algebra. There, extensions δ′ are calculated which generate strongly continuous semigroups of ∗-homomorphisms. These semigroups do not extend to one-parameter groups unless the implementing symmetric operator in one-particle space is already self-adjoint