Infinite Dimensional Ito Algebras of Quantum White Noise

Details Infinite Dimensional Ito Algebras of Quantum White Noise Infinite Dimensional Ito Algebras of Quantum White Noise

2005-12-13

arxiv.org/abs/math/0512288v1

Trends in Contemporary Infinite Dimensional Analysis and Quantum Probability 57--80, Instituto Italiano di Cultura, Kyoto, 200

Mathematics

Probability

15 pages, read at the International Conference on Infinite Dimensional Analysis and Quantum Probability, Kyoto, 2000

Scientific paper

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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