Subject Area: Operator Theory; Operator Algebras
We have studied dilation of Completely Positive (CP) semi-groups on C* or von Neumann algebras, employing quantum stochastic calculus developed by Hudson and Parthasarathy. For a uniformly continuous CP semi-group, there is general way to construct EH flow. However, in case of strongly continuous CP semi-group the generator is unbounded and there is no general method. For a class of strongly continuous CP semi-groups on uniformly hyper-finite (UHF) C*-algebras, we have obtained EH dilations using iteration technique and Random walk approach.
We have made some progress on characterization of unitary processes on Hilbert space. Under certain assumptions we have shown that processes with uniformly ( strongly) continuous expectation semi-groups are unitarily equivalent to a Hudson-Parthasarathy (HP) flow.
Currently, we are exploring possible construction of minimal semi-group of completely positive Maps on C*-algebras or von Neumann algebra from formal generators. As the generator here is unbounded, under certain hypothesis there are some results using Feynman-Kac formula.
We are also trying to understand the class of non-CP maps on Mn(ℂ). Such maps are capable of detecting entangled states.
We are also intrested in spectral analysis and perturbation of self adjoint operators ( not necessarily bounded). Determining various spectrums of a self adjoint operator is often a big challenge. There are some results using integral transforms.