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    Rights statement: The final, definitive version of this article has been published in the Journal, Journal of Mathematical Analysis and Applications 409 (2), 2013, © ELSEVIER.

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    Date added: 14/10/13

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Sesquilinear quantum stochastic analysis in Banach space

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<mark>Journal publication date</mark>15/01/2014
<mark>Journal</mark>Journal of Mathematical Analysis and Applications
Issue number2
Volume409
Number of pages21
Pages (from-to)1031-1051
<mark>State</mark>Published
Early online date19/09/13
<mark>Original language</mark>English

Abstract

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.

Bibliographic note

The final, definitive version of this article has been published in the Journal, Journal of Mathematical Analysis and Applications 409 (2), 2013, © ELSEVIER.