Structural properties of reduced $C^*$-algebras associated with higher-rank lattices

Abstract

We present the first examples of higher-rank lattices whose reduced $C^{*}$-algebras satisfy strict comparison, stable rank one, selflessness, uniqueness of embeddings of the Jiang--Su algebra, and allow explicit computations of the Cuntz semigroup. This resolves a question raised in recent groundbreaking work of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchell, in which they exhibited a large class of finitely generated non-amenable groups satisfying these properties. Our proof relies on quantitative estimates in projective dynamics, crucially using the exponential mixing for diagonalizable flows. As a result, we obtain an effective mixed-identity-freeness property, which, combined with V. Lafforgue's rapid decay theorem, yields the desired conclusions.