Quasifree martingales
Abstract
A noncommutative Kunita-Watanabe-type representation theorem is established for the martingales of quasifree states of CCR algebras. To this end the basic theory of quasifree stochastic integrals is developed using the abstract It\^o integral in symmetric Fock space, whose interaction with the operators of Tomita-Takesaki theory we describe. Our results extend earlier quasifree martingale representation theorems in two ways: the states are no longer assumed to be gauge-invariant, and the multiplicity space may now be infinite-dimensional. The former involves systematic exploitation of Araki's Duality Theorem. The latter requires the development of a transpose on matrices of unbounded operators, defying the lack of complete boundedness of the transpose operation.
Cite
@article{arxiv.1203.6693,
title = {Quasifree martingales},
author = {J. Martin Lindsay and Oliver T. Margetts},
journal= {arXiv preprint arXiv:1203.6693},
year = {2018}
}
Comments
29 pages