The spatial Rokhlin property for actions of compact quantum groups
The spatial Rokhlin property for actions of compact quantum groups
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DOI：
10.1016/j.jfa.2016.09.023
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发表时间：
2016-05
影响因子：
1.7
通讯作者：
Selccuk Barlak;G'abor Szab'o;Christian Voigt
中科院分区：
文献类型：
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作者：
Selccuk Barlak;G'abor Szab'o;Christian Voigt
我们引进了C-⁎-代数上上紧量子群作用的空间罗克林性质，推广了经典紧群作用和有限量子群作用的罗克林性质。我们方法的两个关键部分是顺序分裂⁎-同态的概念，以及使用辫子张量积而不是普通张量积。我们证明了各种结构结果从经典理论延续到这个更一般的设置。特别地，我们证明了与分类程序相关的一些C⁎-代数性质从一个罗克林作用的基础C⁎-代数传递到交叉积和不动点代数。为了建立分类理论，我们证明了Rokhlin作用量相对于近似酉等价性具有刚性性质。在对偶理论方面，我们引入了离散量子群行为的空间近似表象概念。证明了上精确紧量子群的作用的空间Rokhlin性质对于其对偶离散量子群的作用是对偶到空间近似表示的，反之亦然。
We introduce the spatial Rokhlin property for actions of coexact compact quantum groups on C⁎-algebras, generalizing the Rokhlin property for both actions of classical compact groups and finite quantum groups. Two key ingredients in our approach are the concept of sequentially split⁎-homomorphisms, and the use of braided tensor products instead of ordinary tensor products. We show that various structure results carry over from the classical theory to this more general setting. In particular, we show that a number of C⁎-algebraic properties relevant to the classification program pass from the underlying C⁎-algebra of a Rokhlin action to both the crossed product and the fixed point algebra. Towards establishing a classification theory, we show that Rokhlin actions exhibit a rigidity property with respect to approximate unitary equivalence. Regarding duality theory, we introduce the notion of spatial approximate representability for actions of discrete quantum groups. The spatial Rokhlin property for actions of a coexact compact quantum group is shown to be dual to spatial approximate representability for actions of its dual discrete quantum group, and vice versa.