Crossed Products of Operator Algebras: Applications of Takai Duality

Let (G,Σ) be an ordered abelian group with Haar measure μ, let (A,G, α) be a dynamical system and let A oα Σ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism Aoα Σ ∼s ( A⊗K(G,Σ, μ) ) oα⊗Ad ρ G, where K(G,Σ, μ) denotes the compact operators in the CSL algebra AlgL(G,Σ, μ) and ρ denotes the right regular representation of G. We also show that there exists a complete lattice isomorphism between the α̂-invariant ideals of Aoα Σ and the (α⊗Ad ρ)-invariant ideals of A⊗ K(G,Σ, μ). Using Takai duality we also continue our study of the Radical for the crossed product of an operator algebra and we solve open problems stemming from the earlier work of the authors. Among others we show that the crossed product of a radical operator algebra by a compact abelian group is a radical operator algebra. We also show that the permanence of semisimplicity fails for crossed products by R. A final section of the paper is devoted to the study of radically tight dynamical systems, i.e., dynamical systems (A,G, α) for which the identity Rad(Aoα G) = (RadA)oα G persists. A broad class of such dynamical systems is identified.