An Algebra of Non-commutative Bounded Semimartingales: Square and Angle Quantum Brackets

Thanks to the extension of the non-commutative stochastic calculus on Fock space developed by Attal and Meyer, we give a ∗-algebra for the composition of non-commutative semimartingales. We give a characterization of this algebra in terms of the regularity of the semimartingales with respect to some Radon measures. This characterization is an extension for quantum semimartingales of the quantum martingale representation theorem of Parthasarathy and Sinha [J. Funct. Anal.67 (1986), 126-151]. We develop a non-commutative stochastic calculus by defining non-commutative square and angle brackets, which are extensions of the classical ones and verify most of the usual properties. A non-commutative Ito formula for polynomials of non-commutative semimartingales (which also corresponds to the usual one in the commutative case) is obtained. An intrinsic definition of the square bracket is given by proving that it can be interpreted as a quadratic variation of the processes. Finally, all the corresponding results for a Fock space with any, at most countable, multiplicity are given.