\(*\)-regular Leavitt path algebras of arbitrary graphs. (Q353580): Difference between revisions

For a positive integer \(n\), an involution \(*\) on a ring \(R\) is said to be \(n\)-proper if for any \(x_1,\dots,x_n\in R\) one has that \(x_1x_1^*+\cdots+x_nx_n^*=0\) implies \(x_i=0\) for any \(i\). When \(n=1\) we say that the involution is proper. The involution \(*\) is positive-definite if it is \(n\)-proper for every \(n\). If \(K\) is a field with involution and \(E\) an arbitrary graph, the involution of \(K\) induces an involution of the Leavitt path algebra \(L_K(E)\). It is proved in this work that the involution in \(L_K(E)\) is proper if the involution in \(K\) is positive-definite. Furthermore the following three assertions are proved to be equivalent for a field \(K\) with involution: (i) the involution on \(K\) is positive-definite, (ii) The involution on \(L_K(E)\) is positive-definite for every graph \(E\); and (iii) The involution on \(L_K(E)\) is positive-definite for some graph \(E\). In section 4 the authors investigate the question of \(*\)-regularity of Leavitt path algebras. It is known that \(L_K(E)\) is (von Neumann) regular if the graph \(E\) is acyclic. It is proved in this section that for a ring \(R\) with local units, \(R\) is \(*\)-regular if and only if \(R\) is regular and \(*\) is proper. Then the characterization of \(*\)-regularity can be stated in the following terms: let \(K\) be a field with involution and \(E\) an arbitrary graph, the following assertions are equivalent: (i) \(L_K(E)\) is \(*\)-regular, (ii) \(L_K(E)\) is regular and proper, (iii) \(E\) is acyclic and \(K\) is \(n\)-proper for some finite \(n\) (more details on this \(n\) in the paper). As a consequence, it is proved 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 is also true for Leavitt path algebras over arbitrary graphs.