Quantum stop times

Abstract

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(R ₊)) 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.