From Stable Rank One to Real Rank Zero: A Note on Tracial Approximate Oscillation Zero
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911344941334528 |
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| author | Fu, Xuanlong |
| author_facet | Fu, Xuanlong |
| contents | We present a relation between stable rank one and real rank zero via the method of tracial oscillation. Let $A$ be a simple separable $C^*$-algebra of stable rank one. We show that $A$ has tracial approximate oscillation zero and, as a consequence, the tracial sequence algebra $l^\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace-kernel ideal with respect to 2-quasitraces.
We also show that for a $C^*$-algebra $B$ that has non-trivial 2-quasitraces, $B$ has tracial approximate oscillation zero is equivalent to $l^\infty(B)/J_B$ has real rank zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23911 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From Stable Rank One to Real Rank Zero: A Note on Tracial Approximate Oscillation Zero Fu, Xuanlong Operator Algebras We present a relation between stable rank one and real rank zero via the method of tracial oscillation. Let $A$ be a simple separable $C^*$-algebra of stable rank one. We show that $A$ has tracial approximate oscillation zero and, as a consequence, the tracial sequence algebra $l^\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace-kernel ideal with respect to 2-quasitraces. We also show that for a $C^*$-algebra $B$ that has non-trivial 2-quasitraces, $B$ has tracial approximate oscillation zero is equivalent to $l^\infty(B)/J_B$ has real rank zero. |
| title | From Stable Rank One to Real Rank Zero: A Note on Tracial Approximate Oscillation Zero |
| topic | Operator Algebras |
| url | https://arxiv.org/abs/2512.23911 |