Support expansion $\mathrm C^*$-algebras

TL;DR

This paper introduces support expansion $ ext{C}^*$-algebras, a class of operator algebras on $L^2$ spaces that generalize uniform Roe algebras to continuous and quantum settings, revealing a rich and complex structure.

Contribution

It defines support expansion $ ext{C}^*$-algebras in continuous and quantum contexts and analyzes their intricate poset structure, extending classical discrete results.

Findings

Support expansion $ ext{C}^*$-algebras generalize uniform Roe algebras.

The poset of these algebras in $L^2( eal)$ is extremely rich.

Uncountable chains and antichains exist within this poset.

Abstract

We consider operators on $L^2$ spaces that expand the support of vectors in a manner controlled by some constraint function. The primary objects of study are $\mathrm C^*$-algebras that arise from suitable families of constraints, which we call support expansion $\mathrm C^*$-algebras. In the discrete setting, support expansion $\mathrm C^*$-algebras are classical uniform Roe algebras, and the continuous version featured here provides examples of "measurable" or "quantum" uniform Roe algebras as developed in a companion paper. We find that in contrast to the discrete setting, the poset of support expansion $\mathrm C^*$-algebras inside $\mathcal B(L^2(\mathbb R))$ is extremely rich, with uncountable ascending chains, descending chains, and antichains.