Support expansion ${\mathrm{C}}^{\ast }$-algebras

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 ${\mathrm{C}}^{\ast }$-algebras that arise from suitable families of constraints, which we call support expansion ${\mathrm{C}}^{\ast }$-algebras. In the discrete setting, support expansion ${\mathrm{C}}^{\ast }$-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 ${\mathrm{C}}^{\ast }$-algebras inside $\mathcal{B}(L^2(\mathbb{R}))$ is extremely rich, with uncountable ascending chains, descending chains, and antichains.