Extension of the statistical mechanics of equilibrium to noncommutative constraints

A formula similar to the Gibbs canonical and grand canonical ensembles is proven for the ensemble of maximal entropy among those ensembles of common mean values of possibly n o n c o m m u t i n g operators. This is done over a Hilbert space of finite dimension Δ. A partition matrix Π becomes important; the partition function Z=TrΠ displaces Π in the thermodynamics only in the commutative case. Generalization to Δ infinite is discussed informally.