Explicit Baker--Campbell--Hausdorff Radii in Special Banach--Malcev Algebras of Shifts
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918195689947136 |
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| author | Athmouni, Nassim |
| author_facet | Athmouni, Nassim |
| contents | We establish explicit convergence radii for the Baker--Campbell--Hausdorff (BCH) series in special Banach--Malcev algebras of shifts-those embeddable into a Banach alternative algebra. Under the continuity estimate $\|[x,y]\|\leq B\|x\|\|y\|$, the series converges absolutely whenever $B(\|x\|+\|y\|)<1/(4K)$, where $K\geq1$ bounds the absolute BCH coefficients. The constant $1/(4K)$ stems from a Catalan-number majorization and is sharp in the exponential-weight model. We compute $B$ explicitly for operator, exponential, polynomial, damped, and tree-like shift algebras, including the non-Lie split-octonionic (Zorn) algebra ($B=2$, $ρ=1/(8K)$). All results require the speciality assumption; the framework does not apply to general Malcev algebras. Geometrically, $ρ=1/(4KB)$ is the analyticity radius of the induced Moufang loop; numerically, it governs stability of BCH-type integrators. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_06357 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explicit Baker--Campbell--Hausdorff Radii in Special Banach--Malcev Algebras of Shifts Athmouni, Nassim Rings and Algebras Mathematical Physics Differential Geometry Functional Analysis Operator Algebras 17D10, 22E65, 46H70, 65L05 G.1.1; I.1.2; F.2.2 We establish explicit convergence radii for the Baker--Campbell--Hausdorff (BCH) series in special Banach--Malcev algebras of shifts-those embeddable into a Banach alternative algebra. Under the continuity estimate $\|[x,y]\|\leq B\|x\|\|y\|$, the series converges absolutely whenever $B(\|x\|+\|y\|)<1/(4K)$, where $K\geq1$ bounds the absolute BCH coefficients. The constant $1/(4K)$ stems from a Catalan-number majorization and is sharp in the exponential-weight model. We compute $B$ explicitly for operator, exponential, polynomial, damped, and tree-like shift algebras, including the non-Lie split-octonionic (Zorn) algebra ($B=2$, $ρ=1/(8K)$). All results require the speciality assumption; the framework does not apply to general Malcev algebras. Geometrically, $ρ=1/(4KB)$ is the analyticity radius of the induced Moufang loop; numerically, it governs stability of BCH-type integrators. |
| title | Explicit Baker--Campbell--Hausdorff Radii in Special Banach--Malcev Algebras of Shifts |
| topic | Rings and Algebras Mathematical Physics Differential Geometry Functional Analysis Operator Algebras 17D10, 22E65, 46H70, 65L05 G.1.1; I.1.2; F.2.2 |
| url | https://arxiv.org/abs/2511.06357 |