Geodesic
On the Signature of a Path in an Operator Algebra
Symmetry, integrability and geometry: methods and applications, Tome 18 (2022) Cet article a éte moissonné depuis la source Math-Net.Ru

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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: noncommutative probability, operads, duoidal categories.
Mots-clés : signature 
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     author = {Nicolas Gilliers and Carlo Bellingeri},
     title = {On the {Signature} of a {Path} in an {Operator} {Algebra}},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a95/}
}
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Nicolas Gilliers; Carlo Bellingeri. On the Signature of a Path in an Operator Algebra. Symmetry, integrability and geometry: methods and applications, Tome 18 (2022). http://geodesic.mathdoc.fr/item/SIGMA_2022_18_a95/

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