On Spectral Invariant Dense Subalgebras of Uniform Roe Algebras with Subexponential Growth

TL;DR

This paper investigates spectrally invariant subalgebras within uniform Roe algebras for groups with subexponential growth, extending spectral invariance results beyond polynomial growth cases using admissible weights.

Contribution

It constructs and proves spectral invariance of new subalgebras in uniform Roe algebras for groups with subexponential growth, broadening previous spectral invariance frameworks.

Findings

Constructed a class of subalgebras $R^{ obreakinfty}(G)$ for groups with subexponential growth

Proved spectral invariance of these subalgebras in $C_u^*(G)$

Extended spectral invariance results beyond polynomial growth settings

Abstract

In this paper, we study spectrally invariant subalgebras of uniform Roe algebras for discrete groups with subexponential growth. For a group $G$ with subexponential growth and satisfying property $P$, we construct a class of subalgebras $R^{\infty}(G)$. We then prove their spectral invariance in $C_u^*(G)$ through the application of admissible weights. This extends $\ell^2$-norm spectral invariance results beyond polynomial growth settings.