论文标题
交叉产品上的实际光谱三元
Real Spectral Triples on Crossed Products
论文作者
论文摘要
Given a spectral triple on a unital $C^{*}$-algebra $A$ and an equicontinuous action of a discrete group $G$ on $A$, a spectral triple on the reduced crossed product $C^{*}$-algebra $A\rtimes_r G$ was constructed by Hawkins, Skalski, White and Zacharias in [On spectral triples on crossed products arising from等效动作，数学。扫描。 113（2）（2013）262-291]，扩展了Belissard，Marcolli和Reihani在[频谱度量空间上的动力学系统中的构造，Preprint（2010），Arxiv：1008.4617]，通过使用Kasparov产品为Dirac Operator制作Ansatz。假设$ a $上的三重$对于$ g $的操作是均等的，我们表明，$ a \ rtimes_r g $上的三倍对于$ g $的双共同授课是等效的。如果在$ a $上给出了三倍的三重元素，那么我们为两个不等的真实结构提供了triple $ a \ rtimes_rg $的构造。我们根据$ j $的KO-Dimension计算每个实际结构的KO-Dimension，并表明保留了第一阶和第二阶条件。最后，我们在$ a \ rtimes_rg $上表征了三倍的均等定向周期，来自$ a $的三倍的二次方向周期。我们沿着论文表明，我们的构造概括了在非交通性$ 2 $ -torus上概括性光谱三倍的构造。
Given a spectral triple on a unital $C^{*}$-algebra $A$ and an equicontinuous action of a discrete group $G$ on $A$, a spectral triple on the reduced crossed product $C^{*}$-algebra $A\rtimes_r G$ was constructed by Hawkins, Skalski, White and Zacharias in [On spectral triples on crossed products arising from equicontinuous actions, Math. Scand. 113(2) (2013) 262-291], extending the construction by Belissard, Marcolli and Reihani in [Dynamical systems on spectral metric spaces, preprint (2010), arXiv:1008.4617], by using the Kasparov product to make an ansatz for the Dirac operator. Supposing that the triple on $A$ is equivariant for an action of $G$, we show that the triple on $A\rtimes_r G$ is equivariant for the dual coaction of $G$. If moreover an equivariant real structure $J$ is given for the triple on $A$, we give constructions for two inequivalent real structures on the triple $A\rtimes_rG$. We compute the KO-dimension with respect to each real structure in terms of the KO-dimension of $J$ and show that the first and the second order conditions are preserved. Lastly, we characterise an equivariant orientation cycle on the triple on $A\rtimes_rG$ coming from an equivariant orientation cycle on the triple on $A$. We show, along the paper, that our constructions generalize the respective constructions of the equivariant spectral triple on the noncommutative $2$-torus.
