Baer-Type Ring Characterizations of Steinberg Algebras, with Applications to Leavitt Path Algebras
Abstract
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.
Related
- Zhang twists for semigroups and groupoids, with applications to Steinberg algebras
- General linear and Steinberg groups over the Leavitt algebra L𝔽2(1,2)
- On the existence of divisible non-injective modules over Leavitt path algebras
- Growth functions of algebras and an application to Leavitt path algebras
- Cartan inverse semigroups and twisted Steinberg algebras
BibTeX
@misc{ahmadi2026baertype,
title = {{Baer-Type Ring Characterizations of Steinberg Algebras, with Applications to Leavitt Path Algebras}},
author = {Morteza Ahmadi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05862},
eprint = {2610.05862},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05862},
}
RIS
TY - UNPB
TI - Baer-Type Ring Characterizations of Steinberg Algebras, with Applications to Leavitt Path Algebras
AU - Morteza Ahmadi
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.05862
UR - https://arxiv.org/abs/2610.05862
AB - 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.
ER -