Scale Structures and C*-algebras

TL;DR

This paper explores the duality between large and small scale structures on spaces through the lens of C*-algebras, establishing a correspondence that extends to noncommutative settings involving bounded operators.

Contribution

It introduces hybrid scale structures to fully connect scale structures with C*-subalgebras, extending the duality to noncommutative C*-algebras.

Findings

C*-subalgebras induce both small and large scale structures on spaces.

A full duality requires hybrid structures combining small and large scales.

Extension of the duality to noncommutative C*-algebras of bounded operators.

Abstract

The purpose of this paper is to investigate the duality between large scale and small scale. It is done by creating a connection between C*-algebras and scale structures. In the commutative case we consider C*-subalgebras of $C^b(X)$, the C*-algebra of bounded complex-valued functions on $X$. Namely, each C*-subalgebra $\mathscr{C}$ of $C^b(X)$ induces both a small scale structure on $X$ and a large scale structure on $X$. The small scale structure induced on $X$ corresponds (or is analogous) to the restriction of $C^b(h(X))$ to $X$, where $h(X)$ is the Higson compactification. The large scale structure induced on $X$ is a generalization of the $C_0$-coarse structure of N.Wright. Conversely, each small scale structure on $X$ induces a C*-subalgebra of $C^b(X)$ and each large scale structure on $X$ induces a C*-subalgebra of $C^b(X)$. To accomplish the full correspondence between scale structures on $X$ and C*-subalgebras of $C^b(X)$ we need to enhance the scale structures to what we call hybrid structures. In the noncommutative case we consider C*-subalgebras of bounded operators $B(l_2(X))$.