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- W2949217446 abstract "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 $L_K(E).$ We show that the involution on $L_K(E)$ 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 $L_K(E)$ is regular if and only if $E$ is acyclic. We give necessary and sufficient conditions for $L_{K}(E)$ to be $^ast$-regular (i.e. regular with proper involution). This characterization of $^ast$-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 $^ast$-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." @default.
- W2949217446 created "2019-06-27" @default.
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- W2949217446 date "2013-02-02" @default.
- W2949217446 modified "2023-10-01" @default.
- W2949217446 title "Regular Leavitt path algebras of arbitrary graphs" @default.
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