On the Relation between K- and L-Theory of C^*-Algebras

2016/08/09 by Markus Land, Thomas Nikolaus, Land, Markus +1 · 2 citations
Mathematics · #Advanced Operator Algebra Research #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1608.02903

openalex publication_date 2016/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove the existence of a map of spectra τ_A \colon kA → lA between connective topological K-theory and connective algebraic L-theory of a complex C^*-algebra A which is natural in A and compatible with multiplicative structures. We determine its effect on homotopy groups and as a consequence obtain a natural equivalence KA[1/2] → LA[1/2] of periodic K- and L-theory spectra after inverting 2. We show that this equivalence extends to K- and L-theory of real C^*-algebras. Using this we give a comparison between the real Baum-Connes conjecture and the L-theoretic Farrell-Jones conjecture. We conclude that these conjectures are equivalent after inverting 2 if and only if a certain completion conjecture in L-theory is true.

Citations

• A functorial equivariant K-theory spectrum and an equivariant Lefschetz formula

Cited by

• Chern character for torsion-free ample groupoids
• K(1)-local K-theory of Azumaya algebras

Related

• L-theory of C^*-algebras
• The Baum-Connes and the Farrell-Jones Conjectures in K- and L-Theory
• The spectral model of (Real) K-theory
• Banach KK-theory and the Baum-Connes conjecture
• Real K-Theory for C^*-Algebras: Just the Facts