The Jacobson radical for analytic crossed products

Semicrossed products of the disk algebra and the Jacobson radical

We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with zero hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.

Crossed products of operator algebras: applications of Takai duality

Let $(\mathcal G, Σ)$ be an ordered abelian group with Haar measure $μ$, let $(\mathcal A, \mathcal G, α)$ be a dynamical system and let $\mathcal A\rtimes_α Σ$ be the associated semicrossed product. Using Takai duality we establish a stable isomorphism \[ \mathcal A\rtimes_α Σ\sim_{s} \big(\mathcal A \otimes \mathcal K(\mathcal G, Σ, μ)\big)\rtimes_{α\otimes {\rm Ad}\: ρ} \mathcal G, \] where $\mathcal K(\mathcal G, Σ, μ)$ denotes the compact operators in the CSL algebra ${\rm Alg}\:\mathcal L(\mathcal G, Σ, μ)$ and $ρ$ denotes the right regular representation of $\mathcal G$. We also show th…

稳定子群与约化交叉积的简单性

Stabilizer Subgroups and the Simplicity of Reduced Crossed Products

本文研究了可数群在紧空间上的最小作用的约化交叉积的简单性，证明了若约化交叉积简单，则存在一个点其稳定子群的可解交换余部为平凡。进一步，该研究为可数线性群、双曲群等类别的约化交叉积的简单性提供了完整刻画，并回答了Ozawa (2014)提出的问题。

The Bidual of a Radical Operator algebra can be Semisimple

The paper of S. Gulick [Sidney (Denny) L. Gulick, Commutativity and ideals in the biduals of topological algebras, Pacific J. Math 18 No. 1, 1966] contains some good mathematics, but it also contains an error. It claims that for a Banach algebra A, the intersection of the Jacobson radical of A** with A is precisely the radical of A (this is claimed for either of the Arens products on A** - in itself a reasonable claim, because A is always contained in the topological centre of A**, so a fixed a in A lies in the radical of A** with the first Arens product, if and only if it lies in the radical …

交叉积与张量积的不变冯·诺依曼子代数

Invariant von Neumann subalgebras of crossed products and tensor products

该文研究可数离散群保迹作用下交叉积与张量积的不变冯·诺依曼子代数，在特定假设下分类其结构，并构造了自由群的相关保测作用实例。