Algebraic bivariant KK-theory and Leavitt path algebras.

  • Guillermo Cortiñas

    Universidad de Buenos Aires, Argentina
  • Diego Montero

    Universidad de Buenos Aires, Argentina
Algebraic bivariant $K$-theory and Leavitt path algebras. cover
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We investigate to what extent homotopy invariant, excisive and matrix stable homology theories help one distinguish between the Leavitt path algebras L(E)L(E) and L(F)L(F) of graphs EE and FF over a commutative ground ring \ell. We approach this by studying the structure of such algebras under bivariant algebraic KK-theory kkkk, which is the universal homology theory with the properties above. We show that under very mild assumptions on \ell, for a graph EE with finitely many vertices and reduced incidence matrix AEA_E, the structure of L(E)L(E) in kkkk depends only on the groups Coker(IAE)(I-A_E) and Coker(IAEt)(I-A_E^t). We also prove that for Leavitt path algebras, kkkk has several properties similar to those that Kasparov's bivariant KK-theory has for CC^*-graph algebras, including analogues of the Universal coefficient and Künneth theorems of Rosenberg and Schochet.

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Guillermo Cortiñas, Diego Montero, Algebraic bivariant KK-theory and Leavitt path algebras.. J. Noncommut. Geom. 15 (2021), no. 1, pp. 113–146

DOI 10.4171/JNCG/397