Quasifree martingales

TL;DR

This paper develops a noncommutative martingale representation theorem for quasifree states of CCR algebras, extending previous results by removing gauge-invariance assumptions and allowing infinite-dimensional multiplicity spaces.

Contribution

It introduces a comprehensive theory of quasifree stochastic integrals using the abstract Itô integral, expanding the scope of quasifree martingale representation theorems.

Findings

Extended quasifree martingale representation to non-gauge-invariant states

Allowed infinite-dimensional multiplicity spaces in the theory

Developed a transpose operation for unbounded operators

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.