Crossed products of operator algebras: Applications of Takai duality

Abstract

Let (G, Σ) be an ordered abelian group with Haar measure µ, let (A, G, α) be a dynamical system and let A ⋊α Σ be the associated semicrossed product. Using Takai duality we establish a stable isomorphismwhere K(G, Σ, µ) denotes the compact operators in the CSL algebra Alg L(G, Σ, µ) and ρ denotes the right regular representation of G. We also show that there exists a complete lattice isomorphism between thêUsing 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(A ⋊α G) = (Rad A) ⋊α G persists. A broad class of such dynamical systems is identified.

Crossed Products of Operator Algebras

“…This leads to more examples of non-semisimple operator algebras which give semisimple crossed products. In [58] it is also shown…”

“…In [58], the authors of the present paper continue their investigation on Takai duality and its applications. In particular, they show that any semicrossed product of an operator algebra by an abelian ordered group is stably isomorphic to a non-selfadjoint crossed product.…”

The hyperrigidity of tensor algebras of C⁎-correspondences

“…This theory was in turn leveraged to provide a positive confirmation to the Hao-Ng isomorphism problem in the case of graph correspondences and arbitrary groups. For further reading on the subject please see [9,10,11].…”

On the primitive ideals of nest algebras

“…The Jacobson radical has been a frequent object of study in non-self-adjoint algebras, and considerable effort has been expended to identify the radical in the context of various classes of non-self-adjoint algebras, e.g. [4,5,8,9,12,16,22,23]. Why is this?…”