On Quantum Itô Algebras and Their Decompositions

Luigi Accardi, Johan Quaegebeur

V. P. Belavkin

A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Ito algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Ito B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Ito algebras of quantum Brownian and quantum Levy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Ito algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Levy (Poisson) algebra. In particular, every quantum thermal noise 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 .

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V. P. Belavkin

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.

Nicolas Privault

J. Martin Lindsay

A natural counterpart to the Lie–Trotter product formula for norm-continuous, one-parameter semigroups is proved, for the class of quasicontractive quantum stochastic (QS) operator cocycles whose expectation semigroup is norm continuous. Compared to previous such results, the assumption of a strong form of independence of the constituent cocycles is overcome. The analysis is facilitated by the development of some quantum Itô algebra. It is also shown how the maximal Gaussian component of a QS generator may be extracted—leading to a canonical decomposition of such generators, and the connection to perturbation theory is described. Finally, the quantum Itô algebra is extended to quadratic form generators, and a conjecture is formulated for the extension of the product formula to holomorphic QS cocycles.