On some properties of free commutators with semicircular variables
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909908832616448 |
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| author | Popa, Mihai Szpojankowski, Kamil |
| author_facet | Popa, Mihai Szpojankowski, Kamil |
| contents | We investigate commutators of free variables of the form \( i[x, s] \), where \( s \) is a semicircular element. We show that although \( s \) and \( i[x, s] \) are not free, their sum nevertheless satisfies the free additive convolution identity \[ μ_{s + i[x, s]} = μ_s \boxplus μ_{i[x, s]}. \] Furthermore, we prove that the polynomial \( x + i[x, s] \) is freely infinitely divisible whenever \( x \) itself is freely infinitely divisible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13578 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On some properties of free commutators with semicircular variables Popa, Mihai Szpojankowski, Kamil Operator Algebras Combinatorics Functional Analysis 46L54, 05A18 We investigate commutators of free variables of the form \( i[x, s] \), where \( s \) is a semicircular element. We show that although \( s \) and \( i[x, s] \) are not free, their sum nevertheless satisfies the free additive convolution identity \[ μ_{s + i[x, s]} = μ_s \boxplus μ_{i[x, s]}. \] Furthermore, we prove that the polynomial \( x + i[x, s] \) is freely infinitely divisible whenever \( x \) itself is freely infinitely divisible. |
| title | On some properties of free commutators with semicircular variables |
| topic | Operator Algebras Combinatorics Functional Analysis 46L54, 05A18 |
| url | https://arxiv.org/abs/2511.13578 |