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Quantum stochastic calculus on full Fock modules. (English) Zbl 0973.46057

Let \(B\) be a \(C^*\)-algebra and let \(E\) be a two sided Hilbert \(B-B\)-module. The author considers the full Fock space \[ {\mathcal F}(E) = \bigoplus_{n=0}^\infty E^{\odot n} \] over \(E\), and constructs a (free) quantum stochastic calculus on \({\mathcal F}(E)\) in which all stochastic integrals exist as limits of Riemann sums. Conditions of existence, uniqueness, and unitarity are given for the solutions of quantum stochastic differential equations. Solutions of stochastic differential equations with “stationary and independent increments” are shown to be cocycles. Dilations of CP-semigroups with Christensen-Evans generators on arbitrary \(C^*\)-algebras are constructed without requiring the use of infinite degrees of freedom. In the particular case where \(G\) is a Hilbert space and \(B\) is the algebra \({\mathcal B}(G)\) of bounded linear operators on \(G\), the calculus on full Fock module reduces to quantum stochastic calculus on the product of the full Fock space with the initial space \(G\). As an other application it is shown that the calculus on full Fock module can include calculus on Boolean Fock space as a particular case.

MSC:

46L60 Applications of selfadjoint operator algebras to physics
81S25 Quantum stochastic calculus
46L65 Quantizations, deformations for selfadjoint operator algebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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