标准迹与直接有限Leavitt路代数
Canonical traces and directly finite Leavitt path algebras
摘要：
受图$C^*$ -代数上迹研究的启发，我们考虑了图$C^*$ -代数的代数对应物--Leavitt路代数上的迹（可加的，中心映射）.特别是，我们认为迹消失的非零分次分量的莱维特路代数，并把它们称为{ \em规范}，因为它们是唯一确定的，它们的值的顶点。$C^*$ -代数上的$\mathbb{C}$ -值迹的一个理想性质是正锥的元素的迹是非负的。我们适应这一性质的迹的Leavitt路代数$L_K(E)$的值在任何对合环。我们把具有这种性质的迹称为正迹。如果一个正迹在正元素上是单射的，我们说它是忠实的。我们的特点时，一个典型的，$K$ -线性跟踪是积极的，当它是忠实的，在其值的顶点。作为结果，我们得到了可数图$E$的$L_{\mathbb{C}}(E)$上的忠实的、规范不变的、$\mathbb{C}$ -值（代数）迹的集合与相应的图$C^*$ -代数$C^*(E)$上的忠实的、半有限的、下连续的、规范不变的（算子论）迹的集合之间的双射对应.利用单位环适应于局部单位环的直接有限条件（即$xy=1$蕴涵$yx=1$），我们将有限Leavitt路代数直接刻画为具有无圈出口的基础图的路代数.我们的证明涉及考虑有限子图的“局部”Cohn-Leavitt子代数。最后，我们表明，虽然相关的，类的局部noether，类的直接有限的，和类的莱维特路代数，承认一个忠实的跟踪一般是不同的。
英文摘要：
Motivated by the study of traces on graph $C^*$-algebras, we consider traces (additive, central maps) on Leavitt path algebras, the algebraic counterparts of graph $C^*$-algebras. In particular, we consider traces which vanish on nonzero graded components of a Leavitt path algebra and refer to them as {\em canonical} since they are uniquely determined by their values on the vertices. A desirable property of a $\mathbb{C}$-valued trace on a $C^*$-algebra is that the trace of an element of the positive cone is nonnegative. We adapt this property to traces on a Leavitt path algebra $L_K(E)$ with values in any involutive ring. We refer to traces with this property as positive. If a positive trace is injective on positive elements, we say that it is faithful. We characterize when a canonical, $K$-linear trace is positive and when it is faithful in terms of its values on the vertices. As a consequence, we obtain a bijective correspondence between the set of faithful, gauge invariant, $\mathbb{C}$-valued (algebra) traces on $L_{\mathbb{C}}(E)$ of a countable graph $E$ and the set of faithful, semifinite, lower semicontinuous, gauge invariant (operator theory) traces on the corresponding graph $C^*$-algebra $C^*(E)$. With the direct finite condition (i.e $xy=1$ implies $yx=1$) for unital rings adapted to rings with local units, we characterize directly finite Leavitt path algebras as exactly those having the underlying graphs in which no cycle has an exit. Our proof involves consideration of "local" Cohn-Leavitt subalgebras of finite subgraphs. Lastly, we show that, while related, the class of locally noetherian, the class of directly finite, and the class of Leavitt path algebras which admit a faithful trace are different in general.
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