On the Signature of a Path in an Operator Algebra

TL;DR

This paper introduces a new class of operators based on the signature of paths in a $C^{ ext{star}}$ algebra, providing tools for Taylor expansions of solutions to noncommutative differential equations.

Contribution

It defines operators from path signatures in noncommutative algebras and explores their role in representing solutions and trajectories in noncommutative probability.

Findings

Operators form the basis of Taylor expansions for noncommutative differential equations

Construction of a group of representations related to path signatures

Introduction of a noncommutative signature suited for probability theory

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.