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 M_{n}(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.

- (with D. Goswami and K. B. Sinha), Dilation of a class of quantum dynamical semi-groups with unbounded generator on UHF algebras, Ann. Inst. Henri Poincar�, Probabilit�s et Statistiqu�s, 41, (2005), 505-522.
- Quantum random walks and their convergence to Evans-Hudson Flow, Proc. Ind. Acd. Sci. (Math Sci) 118, (2008), 443-465.
- (with D. Goswami), Quantum Random Walks and Vanishing of the Second Hochschild Cohomology, Letters in Mathematical Physics, 84 (2008), 1-14.
- (with U. C. Ji and K. B. Sinha), Characterization of unitary processes with independent increments, Communications on Stochastic Analysis, 4, (2010), 593-614.