THE STRUCTURE OF THE QUANTUM SEMIMARTINGALE ALGEBRAS
Stéphane Attal · 2007
Abstract. In the theory of quantum stochastic calculus one disposes of two quantum semimartingale algebras S and S ′. The first one is an algebra for the composition of operators and has a quantum functional calculus for analytical functions. The second one is larger and is an algebra for the operations of quantum square and angle brackets. In this article we study the algebraic and analytic properties of these algebras. This study is mainly performed through a remarkable transform of quantum processes which, surprisingly, establishes a bijection in between these two algebras. This bijection allows to define norms on these algebras that equip them with Banach algebra structures.