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Sesquilinear quantum stochastic analysis in Banach space

Research output: Contribution to journalJournal article

Published

Journal publication date15/01/2014
JournalJournal of Mathematical Analysis and Applications
Journal number2
Volume409
Number of pages21
Pages1031-1051
Early online date19/09/13
Original languageEnglish

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.