A Case Study in Non-Commutative Topology

TL;DR

This paper provides an accessible overview of the irrational rotation $C^*$-algebra, illustrating its natural occurrences in physics and mathematics, and explaining how $K$-theory helps analyze it, all in an informal, conversational style.

Contribution

It offers an expository, non-technical introduction to non-commutative topology through the example of the irrational rotation algebra, emphasizing intuition and applications.

Findings

Connections to quantum mechanics, group actions, and foliations.

Use of $K$-theory to extract information from the algebra.

Accessible, informal explanation style.

Abstract

This is an expository note focused upon one example, the irrational rotation $C^*$-algebra. We discuss how this algebra arises in nature - in quantum mechanics, group actions, and foliations, and we explain how $K$-theory is used to get information out of it. Our goal is to write as if we are sitting in Starbucks and explaining an idea to a good friend (on napkins, of course). So we are interested in getting an idea across but not at all interested in the technical details that, in any event, would be lost if the coffee spilled. So come with us for a drink at Starbucks!