The ideal property in crossed products

We describe the lattice of the ideals generated by projections and prove a characterization of the ideal property for ``large" classes of crossed products of commutative -algebras by discrete, amenable groups; some applications are also given. We prove that the crossed product of a -algebra with the ideal property by a group with the ideal property may fail to have the ideal property; this answers a question of Shuzhou Wang.

     

A note on maximality of analytic crossed products

Let G be a compact abelian group with the totally ordered dual group which admits the positive semigroup . Let N be a von Neumann algebra and be an automorphism group of on N. We denote to the analytic crossed product determined by N and α. We show that if is a maximal σ-weakly closed subalgebra of , then induces an archimedean order in .     

The ideal property, the projection property, continuous fields and crossed products

Let A be the C^∗-algebra associated to an arbitrary continuous field of C^∗-algebras. We give a necessary and sufficient condition for A to have the ideal property and, if moreover A is separable, we give a necessary and sufficient condition for A to have the projection property. Some applications of these results are given. We also prove that “many” crossed products of commutative C^∗-algebras by discrete, amenable groups have the projection property, generalizing some of our previous results.     

The complexity of crossed products

Let H be a finite-dimensional Hopf algebra, let A be a finite-dimensional algebra measured by H, and let A # _σ H be a crossed product. In this paper, we first show that if H is semisimple as well as its dual H*, then the complexity of A # _σ H is equal to that of A. Furthermore, we prove that the complexity of a finite-dimensional Hopf algebra H is equal to the complexity of the trivial module _H k. As an application, we prove that the complexity of Sweedler’s 4-dimensional Hopf algebra H ₄ is equal to 1.     

The global dimensions of crossed products and crossed coproducts

In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H^＊ are semisimple, then gl.dim（A#σH）=gl.dim（A）, where a is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A^#σH and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim（C α H）=gl.dim（C）.     

Presence or absence of analytic structure in maximal ideal spaces

We study extensions of Wermer’s maximality theorem to several complex variables. We exhibit various smoothly embedded manifolds in complex Euclidean space whose hulls are non-trivial but contain no analytic disks. We answer a question posed by Lee Stout concerning the existence of analytic structure for a uniform algebra whose maximal ideal space is a manifold.     

Hereditary crossed products

We characterize when a crossed product order over a maximal order in a central simple algebra by a finite group is hereditary. We need only concentrate on the cases when the group acts as inner automorphisms and when the group acts as outer automorphisms. When the group acts as inner automorphisms, the classical group algebra result holds for crossed products as well; that is, the crossed product is hereditary if and only if the order of the group is a unit in the ring. When the group is acting as outer automorphisms, every crossed product order is hereditary, regardless of whether the order of the group is a unit in the ring.

     

The maximal ideal space of analytic Lipschitz algebras

LetK be a compact subset of the complex plane and 0<α≤1. In this article we determine the maximal ideal space ofLi _pA(K, α).     

Toeplitz operators associated with analytic crossed products II (invariant subspaces and factorization)

In [16], we introduced the notion of Toeplitz operators associated with analytic crossed products. In this paper, we study the structure of invariant subspaces with respect to the analytic crossed products. We also investigate the inner-outer factorization problems for analytic Toeplitz operators, the factorization problem for non-negative Toeplitz operators and Szegö's infimum problem.This work was supported in part by a Grant-in-Aid for Scientific Research from the Japanese Ministry of Education.     

2- 上循环交叉积

利用 von Neumann 代数 M 和带有正规的2- 上循环 μ 的离散群 G, 定义了2- 上循环交叉积, 推广了经典的离散交叉积, 并证明了2- 上循环交叉积具有结合律.     

Continuous-trace groupoid crossed products

Let be a second countable, locally compact groupoid with Haar system, and let be a bundle of -algebras defined over the unit space of on which acts continuously. We determine conditions under which the associated crossed product is a continuous trace -algebra.

     

On deformations of crossed products

Let A∗Γ be a crossed product algebra, where A is semisimple, finitely generated over its center and Γ is a finite group. We give a necessary and sufficient condition in terms of the outer action of Γ on A for the existence of a multi-parametric semisimple deformation of the form A((t₁,…,t_n))∗Γ (with the induced outer action). The main tool in the proof is the solution of the so-called twisting problem. We also give an example which shows that the condition is not sufficient if one drops the condition on the finite generation of A over its center.