Abstract
Given a conditionally completely positive map \({\mathcal{L}}\) on a unital *-algebra \({\mathcal{A}}\) , we find an interesting connection between the second Hochschild cohomology of \({\mathcal{A}}\) with coefficients in the bimodule \({E_{\mathcal L} = {\mathcal B}^a (\mathcal{A} \oplus M)}\) of adjointable maps, where M is the GNS bimodule of \({\mathcal{L}}\) , and the possibility of constructing a quantum random walk [in the sense of (Attal et al. in Ann Henri Poincar 7(1):59–104, 2006; Lindsay and Parthasarathy in Sankhya Ser A 50(2):151–170, 1988; Sahu in Quantum stochastic Dilation of a class of Quantum dynamical Semigroups and Quantum random walks. Indian Statistical Institute, 2005; Sinha in Banach Center Publ 73:377–390, 2006)] corresponding to \({\mathcal{L}}\) .
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D. Goswami was supported by a project funded by the Indian National Academy of Sciences.
L. Sahu had research support from the National Board of Higher Mathematics, DAE (India) is gratefully acknowledged.
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Goswami, D., Sahu, L. Quantum Random Walks and Vanishing of the Second Hochschild Cohomology. Lett Math Phys 84, 1–14 (2008). https://doi.org/10.1007/s11005-008-0233-z
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DOI: https://doi.org/10.1007/s11005-008-0233-z