Journal of Operator Theory

Volume 46, Issue 2, Fall 2001 pp. 391-410.

The structure of the quantum semimartingale algebras

Authors: Stephane Attal
Author institution: Université de Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, UFR de Mathématiques, B.P. 74, 38402 St. Martin d'Hères Cedex, France

Summary: In the theory of quantum stochastic calculus one disposes of two quantum semimartingale algebras $\scc$ and $\scc'$. 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.

Keywords: Quantum stochastic calculus, quantum semimartingales, quantum brackets, quantum Ito formula

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