Regular Leavitt Path Algebras of Arbitrary Graphs

• Published in: Acta mathematica Sinica. English series Vol. 28; no. 5; pp. 957 - 968
• Main Authors: Aranda Pino, Gonzalo, Rangaswamy, Kulumani, Vaš, Lia
• Format: Journal Article
• Language: English
• Published: Heidelberg Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 01.05.2012; Springer Nature B.V
• Subjects: Algebra; Graph theory; Graphs; Mathematical analysis; Mathematics; Mathematics and Statistics; Rings (mathematics); Studies; Theorems; 代数性质; 充分必要条件; 意图; 电子领域; 莱维; 路代数; 路径代数; 非周期性; 16S99; arbitrary graph; 16D70; Leavitt path algebra; involution; 16W10; regular
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-011-0106-8

If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra LK（E）. We show that the involution on LK（E） is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra LK（E） is regular if and only if E is acyclic. We give necessary and sufficient conditions for LK（E） to be ＊-regular （i.e., regular with proper involution）. This characterization of ＊-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graph- theoretic properties of E alone. As a corollary, we show that Handelman＇s conjecture （stating that every ,-regular ring is unit-regular） holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.