On some topics of analysis on noncommutative spaces

TL;DR

This paper develops a differential calculus framework for noncommutative spaces modeled by semi-finite W*-algebras, introducing a Riemannian metric analogue and analyzing related PDEs.

Contribution

It constructs a noncommutative differential structure, including a Riemannian metric analogue, and establishes existence and uniqueness results for associated PDEs.

Findings

Constructed a noncommutative Riemannian metric analogue

Formulated a Poincaré-type inequality in the noncommutative setting

Proved existence and uniqueness of solutions for noncommutative PDEs

Abstract

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.