Viacheslav Belavkin (School of Mathematics, University of Nottingham, UK)

Quantum Lévy-Itô fields over modular algebras

Abstract:
      An algebraic characterization of the general (infinite dimensional, noncommutative) Itô algebra is given and a fundamental matrix realization for such algebras in the Krein space and its exponential in Fock space is described. The notion of four-normed Itô B*-algebra, generalizing the C*-algebra is introduced to include the Banach infinite dimensional Itô algebras of noncommutative Brownian and Lévy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. The first ones posses a vacuum vector in the Krein space, the second ones are described by a generalized Hilbert modular algebra
      It is proved that every Itô algebra is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular, every quantum thermal field is the orthogonal sum of a quantum Gaussian field and a quantum Poisson field as it is stated by the Lévy-Khinchin theorem in the classical stochastic case corresponding to the commutative Itô algebras.The well known Lévy-Khinchin classification of the classical noise can be reformulated in purely algebraic terms as the decomposability of any commutative Itô algebra into Wiener (Brownian) and Poisson (Lévy) orthogonal components. In the general case we shall show that every Itô ⋆-algebra is also decomposable into a quantum Brownian, and a quantum Lévy orthogonal components.
      Thus classical stochastic calculus developed by Itô, and its quantum stochastic analog, given by Hudson and Parthasarathy in 2, has been unified in a ⋆-algebraic approach to the operator integration in Fock space 3, in which the classical and quantum calculi become represented as two extreme commutative and noncommutative cases of a generalized Itô calculus.