### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 096, 43 pages      arXiv:2102.11816      https://doi.org/10.3842/SIGMA.2022.096
Contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Action

### On the Signature of a Path in an Operator Algebra

Nicolas Gilliers a and Carlo Bellingeri b
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

Received January 11, 2022, in final form November 30, 2022; Published online December 09, 2022

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.

Key words: signature; noncommutative probability; operads; duoidal categories.

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