Symmetry Integrability and Geometry-Methods and Applications | |
On the Signature of a Path in an Operator Algebra | |
article | |
Nicolas Gilliers1  Carlo Bellingeri2  | |
[1] Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse;Technische Universität Berlin | |
关键词: signature; noncommutative probability; operads; duoidal categories.; | |
DOI : 10.3842/SIGMA.2022.096 | |
来源: National Academy of Science of Ukraine | |
【 摘 要 】
We introduce a class of operators associated with the signature of a smooth path $X$ with values in a $C^{\star}$ algebra $\mathcal{A}$. These operators serve as the basis of Taylor expansions of solutions to controlled differential equations of interest in noncommutative probability. They are defined by fully contracting iterated integrals of $X$, seen as tensors, with the product of $\mathcal{A}$. Were it considered that partial contractions should be included, we explain how these operators yield a trajectory on a group of representations of a combinatorial Hopf monoid. To clarify the role of partial contractions, we build an alternative group-valued trajectory whose increments embody full-contractions operators alone. We obtain therefore a notion of signature, which seems more appropriate for noncommutative probability.
【 授权许可】
Unknown
【 预 览 】
Files | Size | Format | View |
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RO202307120000517ZK.pdf | 2614KB | download |