Boundary operator algebras for free uniform tree lattices

Abstract

Let $X$ be a finite connected graph, each of whose vertices has degree at least three. The fundamental group $\Gamma$ of $X$ is a free group and acts on the universal covering tree $\Delta$ and on its boundary $\partial \Delta$, endowed with a natural topology and Borel measure. The crossed product $C^{\ast }$-algebra $C(\partial \Delta )⋊\Gamma$ depends only on the rank of $\Gamma$ and is a Cuntz-Krieger algebra whose structure is explicitly determined. The crossed product von Neumann algebra does not possess this rigidity. If $X$ is homogeneous of degree $q+1$ then the von Neumann algebra $L^{\infty }(\partial \Delta )⋊\Gamma$ is the hyperfinite factor of type $II{I}_{\lambda }$ where $\lambda =1/q^2$ if $X$ is bipartite, and $\lambda =1/q$ otherwise.

Cited by in corpus (10)

• Euler characteristics and Gysin sequences for group actions on boundaries
• C*-algebras associated to graphs of groups
• On the K-theory of graph C*-algebras
• K-theory of locally finite graph $C^{\ast }$-algebras
• Boundary $C^{\ast }$-algebras for acylindrical groups
• Graphs, spectral triples and Dirac zeta functions
• Torsion in Boundary Coinvariants and K-theory for Affine Buildings
• Graph Reconstruction and Quantum Statistical Mechanics
• Harmonic cochains and K-theory for ${\tilde{A}}_2$ groups
• Invariant boundary distributions for finite graphs