The K-theory of boundary C*-algebras of symmetric spaces

TL;DR

This work computes the K-theory of boundary C*-algebras $C(G/P_0)\rtimes_r\Gamma$ associated with lattice actions on the maximal Furstenberg boundary of symmetric spaces. It establishes a KK-equivalence up to a dimension shift to algebras of the form $C_0(\Gamma\backslash G/M)$ and, in rank 1, links these algebras to the unit tangent bundle $T^1(\Gamma\backslash X)$, relaxing Spin$^\mathbb{C}$ assumptions. Using this KK-bridge, the authors derive explicit K-theory descriptions in torsion-free cohomology cases via the unit tangent bundle cohomology and the Atiyah–Hirzebruch spectral sequence, and they discuss how Euler characteristics govern torsion in K-theory. The results yield examples where boundary C*-algebras are isomorphic despite non-isometric underlying spaces, illustrating the subtle geometric information retained by boundary algebras and offering classification insights via Kirchberg–Phillips theory. Overall, the paper extends known computations, provides a unified framework across dimensions, and demonstrates how boundary data can classify or distinguish locally symmetric spaces in concrete cases.

Abstract

We compute the K-theory of a collection of C*-algebras, which we refer to as boundary C*-algebras, arising as the crossed product C*-algebras of lattice actions on the maximal Furstenberg boundaries of symmetric spaces of noncompact type. As a result, we add new examples to the collection of known isomorphic boundary C*-algebras which are not spatially isomorphic.

Theorems & Definitions (28)

• Definition 3.1: Parabolic subgroup
• Definition 3.2: Geodesic and Furstenberg boundary
• Proposition 3.3: a_borel_compactifications_2006
• Definition 3.4: Boundary C*-algebras
• Theorem 3.5: Mostow
• Lemma 4.1
• proof
• Proposition 4.2: anantharaman-delaroche_purely_1997
• Proposition 4.3
• proof