Quantum stop times

The notion of stop-time can be naturally translated in a quantum probabilistic framework and this problem has been studied by several authors [1], [2], [3], [4], [5]. Recently Parthasarathy and Sinha [4] have established a factorization property of the $L^2$-space over the Wiener space (regarded as the Fock space over $L^2(\bgroup \bf R \egroup _+) $ ) based on the notion of quantum stop time which is a quantum probabilistic analogue of the strong Markov property. In this note we prove a stronger result which has no classical analogue namely that the algebra generated by the stopped Weyl operators in the sense of [4] (i.e.the past algebra with respect to a stop time S), is the algebra of all the bounded operators on $L^2$ of the Wiener space.