Delocalized Eta Invariants, Algebraicity, and K-Theory of Group C*-Algebras Academic Article uri icon


  • Abstract In this paper, we establish a precise connection between higher rho invariants and delocalized eta invariants. Given an element in a discrete group, if its conjugacy class has polynomial growth, then there is a natural trace map on the $K_0$-group of its group $C^ast$-algebra. For each such trace map, we construct a determinant map on secondary higher invariants. We show that, under the evaluation of this determinant map, the image of a higher rho invariant is precisely the corresponding delocalized eta invariant of Lott. As a consequence, we show that if the BaumConnes conjecture holds for a group, then Lotts delocalized eta invariants take values in algebraic numbers. We also generalize Lotts delocalized eta invariant to the case where the corresponding conjugacy class does not have polynomial growth, provided that the strong Novikov conjecture holds for the group.

published proceedings

  • International Mathematics Research Notices

author list (cited authors)

  • Xie, Z., & Yu, G.

citation count

  • 10

complete list of authors

  • Xie, Zhizhang||Yu, Guoliang

publication date

  • July 2021