Structural properties of reduced $C^*$-algebras associated with higher-rank lattices
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912628151943168 |
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| author | Vigdorovich, Itamar |
| author_facet | Vigdorovich, Itamar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_12737 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Structural properties of reduced $C^*$-algebras associated with higher-rank lattices Vigdorovich, Itamar Operator Algebras Dynamical Systems Group Theory 46L35, 46L05, 22D25, 22E40, 20F69 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. |
| title | Structural properties of reduced $C^*$-algebras associated with higher-rank lattices |
| topic | Operator Algebras Dynamical Systems Group Theory 46L35, 46L05, 22D25, 22E40, 20F69 |
| url | https://arxiv.org/abs/2503.12737 |