A finitary criterion for selfless tracial C*-algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917554689146880 |
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| author | Jabbari, Ali |
| author_facet | Jabbari, Ali |
| contents | We study the class of selfless C*-probability spaces introduced by Robert. It is known that a selfless tracial algebra has strict comparison and a unique trace. We prove that for separable tracial C*-algebras, selflessness is equivalent to approximate selflessness, a finitary condition: for every finite set $F$, every $N \geq 1$ and $\varepsilon > 0$ there exists a unitary $u$ with $|τ(u^k)| < \varepsilon$ ($1 \leq |k| \leq N$) and $|τ(w)| < \varepsilon$ for all alternating words $w$ of length $\leq N$ built from centered elements of $F$ and powers $u^n$ ($|n| \leq N$). The equivalence is established using a diagonalisation argument in the tracial ultrapower. As an application, we give a concise proof that countable groups with a topologically-free extreme boundary are C*-selfless. We also discuss the relation to nuclearity and $\mathcal{Z}$-stability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_25382 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A finitary criterion for selfless tracial C*-algebras Jabbari, Ali Operator Algebras 46L05, 46L10, 46L53, 46L54, 03C20, 46B08 We study the class of selfless C*-probability spaces introduced by Robert. It is known that a selfless tracial algebra has strict comparison and a unique trace. We prove that for separable tracial C*-algebras, selflessness is equivalent to approximate selflessness, a finitary condition: for every finite set $F$, every $N \geq 1$ and $\varepsilon > 0$ there exists a unitary $u$ with $|τ(u^k)| < \varepsilon$ ($1 \leq |k| \leq N$) and $|τ(w)| < \varepsilon$ for all alternating words $w$ of length $\leq N$ built from centered elements of $F$ and powers $u^n$ ($|n| \leq N$). The equivalence is established using a diagonalisation argument in the tracial ultrapower. As an application, we give a concise proof that countable groups with a topologically-free extreme boundary are C*-selfless. We also discuss the relation to nuclearity and $\mathcal{Z}$-stability. |
| title | A finitary criterion for selfless tracial C*-algebras |
| topic | Operator Algebras 46L05, 46L10, 46L53, 46L54, 03C20, 46B08 |
| url | https://arxiv.org/abs/2604.25382 |