Strictly positive and strongly positive semigroups

Abstract

We examine positive semigroups acting on Banach lattices and operator algebras. In the lattice framework we characterize strict positivity and strict ordering of holomorphic semigroups by irreducibility criteria. In the algebraic setting we derive ergodic criteria for irreducibility and discuss various aspects of strict positivity. Finally we examine invariant states of a C*-dynamical system in which the automorphism group is replaced by a strongly positive semigroup. We demonstrate that ergodic states are characterized by a cluster property despite the absence of a covariant implementation law for the semigroup.

Strongly positive semigroups and faithful invariant states

“…The equivalences Io3o4 have previously been given by Frigerio [2] with some extraneous continuity assumptions on τ. Frigerio also discusses other criteria of irreducibility. A similar result with faithfulness of ω replaced by a condition of asymptotic abelianness has been given in [5].…”