On the Signature of a Path in an Operator Algebra

Abstract

We introduce a class of operators associated with the signature of a smooth path 𝑋 with values in a 𝐶⋆ algebra 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 𝒜. If it is 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-contraction operators alone. We obtain, therefore, a notion of signature, which seems more appropriate for noncommutative probability.