Morse Theory for C*-Algebras: A Geometric Interpretation of Some Noncommutative Manifolds

TL;DR

This paper extends Morse theory to unital C*-algebras, offering a geometric framework to interpret noncommutative CW complexes and classify unital C*-algebras through modified Morse functions.

Contribution

It introduces a modified Morse theory for unital C*-algebras, enabling geometric interpretation and classification of noncommutative CW complexes.

Findings

Development of tools for geometric interpretation of noncommutative CW complexes

Classification scheme for unital C*-algebras using noncommutative Morse functions

Illustrative examples demonstrating the geometric insights

Abstract

The approach we present is a modification of the Morse theory for unital C*-algebras. We provide tools for the geometric interpretation of noncommutative CW complexes. These objects were introduced and studied in [2],[7] and [14]. Some examples to illustrate these geometric information in practice are given. A classification of unital C*-algebras by noncommutative CW complexes and the modified Morse functions on them is the main object of this work.