Regular Leavitt Path Algebras of Arbitrary Graphs
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 o...
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Published in | Acta mathematica Sinica. English series Vol. 28; no. 5; pp. 957 - 968 |
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Main Authors | , , |
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 | |
Online Access | Get full text |
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Summary: | 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. |
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Bibliography: | Leavitt path algebra, *-regular, involution, arbitrary graph 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. 11-2039/O1 ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
ISSN: | 1439-8516 1439-7617 |
DOI: | 10.1007/s10114-011-0106-8 |