Quantum Random Walks on UHF algebras

In the Ph. D. thesis (S-2005), the problem of obtaining Evans-Hudson dilation for strongly continuous quantum dynamical semigroups on
uniformly hyperfinite C*-algebras is investigated and answered partialy. Here we will present the path followed in the last part of
the thesis. Following the work of Lindsay and Parthasarathy (LP-1988) and Attal and Pautrat (AP-2002), the idea is to construct Evans-Hudson flows as a limit of *-homomorphic families parametrize by partitions of positive real line, using toy Fock space language.
These families of *-homomorphisms are called quantum random walks.
For a class of quantum dynamical semigroup, starting from the unbounded generator, we have
constructed quantum random walks and proved that as the width of the partitions goes to zero the quantum random walks converges weakly and
the limit satisfies a quantum stochastic differential equation. As weak limit of
*-homomorphisms the limit is a completely positive flow but strong conergence of the quantumm random walks is not yet proved and homomorohic property of the limit is unknown.
[AP-2002] Attal S. ; Pautrat, Y. : From repeated to continuous quantum interactions, xxx.lanl.gov/math-ph/0311002.