On some topics of analysis on noncommutative spaces
Danila Zaev

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 -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.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Banach Space Theory · Holomorphic and Operator Theory
