Noncommutative potential theory

We propose to view hermitian metrics on trivial holomorphic vector bundles $E\to \mathrm{\Omega }$ as noncommutative analogs of functions defined on the base $\mathrm{\Omega }$, and curvature as the notion corresponding to the Laplace operator or $\mathrm{\partial }\bar{\mathrm{\partial }}$. We discuss noncommutative generalizations of basic results of ordinary potential theory, mean value properties, maximum principle, Harnack inequality, and the solvability of Dirichlet problems.