Quasifree martingales

Details Quasifree martingales Quasifree martingales

2012-03-30

arxiv.org/abs/1203.6693v1

Mathematics

Operator Algebras

29 pages

Scientific paper

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.

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