On Quantum Ito Algebras and their decompositions

Abstract

A simple axiomatic characterization of the noncommutative Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Ito algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Levy-Khinchin theorem in the classical case. Two basic examples of non-commutative Ito finite group algebras are considered.