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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2022 Volume 18, 096, 43 pp. (Mi sigma1892)

On the Signature of a Path in an Operator Algebra

Nicolas Gilliersa, Carlo Bellingerib

a Institut de Mathématiques de Toulouse, UMR5219, Université de Toulouse, CNRS, UPS, F-31062 Toulouse, France
b Technische Universität Berlin, Straße des 17. Juni 135, 10623 Berlin, Germany

Abstract: 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.

Keywords: signature, noncommutative probability, operads, duoidal categories.

MSC: 18M60, 18M80, 60L10, 46L89

Received: January 11, 2022; in final form November 30, 2022; Published online December 9, 2022

Language: English

DOI: 10.3842/SIGMA.2022.096



Bibliographic databases:
ArXiv: 2102.11816


© Steklov Math. Inst. of RAS, 2025