# Quantum Itô B*-algebras, their classification and decomposition

Banach Center Publications (1998)

- Volume: 43, Issue: 1, page 63-70
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topBelavkin, V.. "Quantum Itô B*-algebras, their classification and decomposition." Banach Center Publications 43.1 (1998): 63-70. <http://eudml.org/doc/208865>.

@article{Belavkin1998,

abstract = {A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Itô B*-algebra, generalizing the C*-algebra, is defined to include the Banach infinite dimensional Itô algebras of quantum Brownian and quantum Lévy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. 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 noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Lévy-Khinchin Theorem in the classical case.},

author = {Belavkin, V.},

journal = {Banach Center Publications},

keywords = {Itô algebra; pseudo-Euclidean fundamental representation; quantum Brownian and quantum Lévy motion; thermal quantum noise; Lévy-Khinchin Theorem},

language = {eng},

number = {1},

pages = {63-70},

title = {Quantum Itô B*-algebras, their classification and decomposition},

url = {http://eudml.org/doc/208865},

volume = {43},

year = {1998},

}

TY - JOUR

AU - Belavkin, V.

TI - Quantum Itô B*-algebras, their classification and decomposition

JO - Banach Center Publications

PY - 1998

VL - 43

IS - 1

SP - 63

EP - 70

AB - A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Itô B*-algebra, generalizing the C*-algebra, is defined to include the Banach infinite dimensional Itô algebras of quantum Brownian and quantum Lévy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. 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 noise is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Lévy-Khinchin Theorem in the classical case.

LA - eng

KW - Itô algebra; pseudo-Euclidean fundamental representation; quantum Brownian and quantum Lévy motion; thermal quantum noise; Lévy-Khinchin Theorem

UR - http://eudml.org/doc/208865

ER -

## References

top- [1] R. L. Hudson, and K. R. Parthasarathy, Quantum Itô's Formula and Stochastic Evolution, Commun. Math. Phys. 93 (1984), 301-323. Zbl0546.60058
- [2] V. P. Belavkin, A new Form and *-algebraic Structure of Quantum Stochastic Integrals in Fock Space, Rendiconti del Seminario Matematico e Fisico di Milano, Vol LVIII (1988), 177-193. Zbl0708.60048
- [3] V. P. Belavkin, Chaotic States and Stochastic Integration in Quantum Systems, Russian Math. Surveys 47(1) (1992), 47-106. Zbl0821.46093
- [4] V. P. Belavkin, Representations of *-semigroups Associated with Infinitely Divisible States, Quantum Probability and Related Topics, Vol VII (1992), 31-50. Zbl0787.46050
- [5] M. Takesaki, J. Funct. Anal. 9 (1972), 306.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.