A note on the isomorphism conjectures for Leavitt path algebras. (Q372655)

Two conjectures related to classification of Leavitt path algebras have been posed in the literature on the subject:NEWLINENEWLINE (1) If \(E\) and \(F\) are finite graphs such that \(L(E)\) and \(L(F)\) are purely infinite simple, then NEWLINE\[NEWLINE(K_0(L(E)),[L(E)])\cong (K_0(L(F)),[L(F)])NEWLINE\]NEWLINE if and only if \(L(E)\cong L(F)\). This was first conjectured in a work by G. Abrams, P. N. Ánh, A. Louly and E. Pardo where an affirmative answer is given in the particular case of graphs whose Leavitt path algebras are purely infinite simple, with at most three vertices and without parallel edges. Then, in another work by G. Abrams, A. Louly, E. Pardo and C. Smith, it is proved that if \(E\) and \(F\) are finite with \(L(E)\) and \(L(F)\) purely infinite simple and Morita equivalent, then (1) is true.NEWLINENEWLINE (2) Let \(E\) and \(F\) be finite graphs, then there is an order preserving \(\mathbb Z[x,x^{-1}]\)-module isomorphism NEWLINE\[NEWLINE(K_0^{\text{gr}}(L(E)),[L(E)])\cong (K_0^{\text{gr}}(L(F)),[L(F)])NEWLINE\]NEWLINE if and only if \(L(E)\cong_{\text{gr}} L(F)\). This conjecture has been formulated in a recent paper by the author in Math. Ann.NEWLINENEWLINE In this paper the author proves that (2) implies (1). On the other hand, for a finite graph \(E\), the canonical \(\mathbb Z\)-grading of the Leavitt path algebra \(L(E)\) is a strong grading if and only if \(E\) has no sinks (this has been proved also in a recent paper by the author and will appear in Isr. J. Math.). In the work under review the author provides a new proof of this fact, using corner skew Laurent polynomial rings.