Quantum random walks and vanishing of the second Hochschild cohomology

Abstract

Given a conditionally completely positive map $\mathcal{L}$ on a unital $\ast$-algebra $\A$, we find an interesting connection between the second Hochschild cohomology of $\A$ with coefficients in the bimodule $E_{\mathcal L}=\B^a(\A \oplus M)$ 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 \cite{AP,LP,L,KBS}) corresponding to $\mathcal{L}$.