THE STRUCTURE OF THE QUANTUM SEMIMARTINGALE ALGEBRAS

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 o...

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Bibliographic Details
Published in:Journal of operator theory 2001-09, Vol.46 (2), p.391-410
Main Author: ATTAL, STÉPHANE
Format: Article
Language:English
Subjects:
Adjoints
Algebra
Integration by parts
Mathematical integrals
Mathematical theorems
Mathematical vectors
Stochastic processes
Tensors
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Summary: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.
ISSN:0379-4024
1841-7744