  We consider a conservative Markov semigroup on a semi-finite $W^*$-algebra.
It is known that under some reasonable assumptions it is enough to determine a
kind of differential structure on such a 'noncommutative space'. We construct
an analogue of a Riemannian metric in this setting, formulate a Poincar\'e-type
inequality, provide existence and uniqueness results for quasi-linear elliptic
and parabolic PDEs defined in terms of the constructed noncommutative calculus.
