发表机构
Tarbiat Modares University（塔里亚特莫达雷斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文推广Baer、Rickart及其局部、分次和$*$-类似物理论到Steinberg代数，证明局部Baer行为迫使群胚离散，并利用轨道分解给出完整刻画，进而应用于Leavitt路代数获得相应环性质刻画。
AI 中文摘要
我们将Baer、Rickart及其局部、分次和$*$-类似物的理论推广到Steinberg代数。首先，我们证明正定性从系数域传递到Steinberg代数，并且这连同由Steinberg和van Wyk刻画的分次von Neumann正则性，产生分次局部Rickart $*$-Steinberg代数。然后我们证明局部Baer行为迫使单位空间的每个紧开子集都是极不连通的。因此，当单位空间的紧开子空间是可度量的时，局部Baer行为迫使群胚是离散的。在离散情形下，我们利用各向同性群代数上的有限矩阵代数的轨道逐轨道分解，将零化子条件转化为显式矩阵条件。对于各向同性平凡或无限循环的群胚，这些要素产生完整的局部和单位刻画。特别地，局部Baer和分次局部Baer条件等价于离散性，而未分次的局部Baer $*$-条件还要求每个具有非平凡各向同性的轨道都是单元素集。然后我们将该理论应用于边界路径群胚。利用Steinberg代数模型，我们获得Leavitt路代数的Rickart、Baer和Baer $*$刻画，包括局部和分次变体。
英文摘要
We extend the theory of Baer, Rickart, and their local, graded, and $*$-analogues for Steinberg algebras. First, we prove that positive definiteness passes from the coefficient field to the Steinberg algebra, and that this, together with graded von Neumann regularity, characterized by Steinberg and van Wyk, yields graded locally Rickart $*$-Steinberg algebras. Then we show that local Baer behavior forces every compact open subset of the unit space to be extremally disconnected. Consequently, when compact open subspaces of the unit space are metrizable, local Baer behavior forces the groupoid to be discrete. In the discrete case, we use the orbit-by-orbit decomposition into finitary matrix algebras over isotropy group algebras to convert annihilator conditions into explicit matrix conditions. For groupoids whose isotropy is trivial or infinite cyclic, these ingredients yield complete local and unital characterizations. In particular, local Baer and graded local Baer conditions are equivalent to discreteness, while the ungraded local Baer $*$-condition additionally requires that every orbit with nontrivial isotropy be a singleton. We then apply the theory to boundary path groupoids. Using the Steinberg algebra model, we obtain the Rickart, Baer, and Baer $*$ characterizations of Leavitt path algebras, including the local and graded variants.
