Abstract
Let G be a group and \(\ell \) a commutative unital \(*\)-ring with an element \(\lambda \in \ell \) such that \(\lambda + \lambda ^*= 1\). We introduce variants of hermitian bivariant K-theory for \(*\)-algebras equipped with a G-action or a G-grading. For any graph E with finitely many vertices and any weight function \(\omega :E^1 \rightarrow G\), a distinguished triangle for \(L(E)=L_\ell (E)\) in the hermitian G-graded bivariant K-theory category \(kk^h_{G_{{\text {gr}}}}\) is obtained, describing L(E) as a cone of a matrix with coefficients in \(\mathbb {Z}[G]\) associated to the incidence matrix of E and the weight \(\omega \). In the particular case of the standard \(\mathbb {Z}\)-grading, and under mild assumptions on \(\ell \), we show that the isomorphism class of L(E) in \(kk^h_{\mathbb {Z}_{{\text {gr}}}}\) is determined by the graded Bowen–Franks module of E. We also obtain results for the graded and hermitian graded K-theory of \(*\)-algebras in general and Leavitt path algebras in particular which are of independent interest, including hermitian and bivariant versions of Dade’s theorem and of Van den Bergh’s exact sequence relating graded and ungraded K-theory.
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Funding for this article came from the sources stated on the title page, and include the University of Buenos Aires, Argentine government agencies CONICET and Agencia Nacional de Promoción Científica y Tecnológica, and the Spanish Ministerio de Ciencia e Innovación.
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Guido Arnone and Guillermo Cortiñas were partially supported by grant UBACyT 256BA from Universidad de Buenos Aires, PIP 2021-2023 GI 11220200100423CO from CONICET and PICT 2017-1395 from Agencia Nacional de Promoción Científica y Técnica. The first named author was supported by PhD fellowships, first from Universidad de Buenos Aires and then from CONICET. The second named author was supported by CONICET and partially supported by grant PGC2018-096446-B-C21 from the Spanish Ministerio de Ciencia e Innovación.
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Arnone, G., Cortiñas, G. Graded K-theory and Leavitt path algebras. J Algebr Comb 58, 399–434 (2023). https://doi.org/10.1007/s10801-022-01184-5
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DOI: https://doi.org/10.1007/s10801-022-01184-5