2018/05/19 by Zhizhang Xie, Guoliang Yu, Xie, Zhizhang +1 · 17 citations
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Combinatorics #Conjecture #Conjugacy class #Delocalized electron #FOS: Mathematics #Group (periodic table) #Invariant (physics) #Invariant theory #K-Theory and Homology (math.KT) #Mathematical analysis #Mathematical physics #Mathematics #Operator Algebras (math.OA) #Physics #Polynomial #Pure mathematics #Quantum mechanics #math.KT #math.OA
paper · pdf · doi:10.48550/arxiv.1805.07617
published in arXiv (Cornell University) (Cornell University) · 27 pages. arXiv admin note: text overlap with arXiv:1804.09026 by other authors
openalex publication_date 2018/05/19 · openalex created_date 2018/06/01 · arxiv created 2019/05/09 · arxiv updated 2019/05/13 · openalex updated_date 2026/07/28
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 K0-group of its group C^∗-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 Baum-Connes conjecture holds for a group, then Lott's delocalized eta invariants take values in algebraic numbers. We also generalize Lott's 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.