Nonpure (Non)Commutative Analysis, Geometry and Mechanics, part 1: differential and integral calculus

TL;DR

This paper develops differential and integral calculus on the space of states of C*-algebras, introducing a formal smooth structure and proving Stokes' theorem in both commutative and noncommutative cases.

Contribution

It constructs a formal smooth calculus on C*-algebra state spaces, extending classical differential geometry to noncommutative settings.

Findings

Established a smooth structure on state spaces of C*-algebras

Proved Stokes' theorem in noncommutative Wasserstein space

Extended classical calculus results to noncommutative geometry

Abstract

We construct and study differential and integral calculus on the space of states of a C*-algebra by equipping it with a formal smooth structure. To achieve this goal we first concentrate on the space of nonpure states of a commutative C*-algebra as a guideline for the noncommutative case. In particular, we prove Stokes' theorem over the both commutative and noncommutative smooth Wasserstein space.