On the topological ranks of Banach $^*$-algebras associated with groups of subexponential growth
Felipe I. Flores
TL;DR
The paper addresses the problem of determining topological stable rank $tsr$ and real rank $rr$ for Banach $^*$-algebras that lie between the Fell-bundle convolution algebra $\ell^1(G\,|\,\mathscr C)$ and its $C^*$-envelope ${\rm C^*}(G\,|\,\mathscr C)$ when $G$ has subexponential growth. The central approach proves that $tsr$ and $rr$ are invariant across all such intermediate algebras, i.e., ${\rm tsr}(\frak B)={\rm tsr}(\ell^1(G\,|\,\mathscr C))$ and ${\rm rr}(\frak B)={\rm rr}(\ell^1(G\,|\,\mathscr C))$, leveraging positive-definite multipliers and spectral invariance, together with unitization arguments. The results are then extended to symmetrized twisted $L^p$-crossed products, enabling explicit calculations of $tsr$ and $rr$ in several dynamical settings, including abelian and nilpotent groups as well as actions on Cantor sets, tori, and Stone–Čech compactifications; in particular, many cases yield topological stable rank $1$. This framework unifies and extends known $C^*$-counterparts to a broad class of Banach $^*$-algebras and $L^p$-crossed-product constructions, providing practical rank data for a range of non-$C^*$-settings.
Abstract
Let $G$ be a group of subexponential growth and $\mathscr C\overset{q}{\to}G$ a Fell bundle. We show that any Banach $^*$-algebra that sits between the associated $\ell^1$-algebra $\ell^1( G\,\vert\,\mathscr C)$ and its $C^*$-envelope has the same topological stable rank and real rank as $\ell^1( G\,\vert\,\mathscr C)$. We apply this result to compute the topological stable rank and real rank of various classes of symmetrized twisted $L^p$-crossed products and show that some twisted $L^p$-crossed products have topological stable rank 1. Our results are new even in the case of (untwisted) group algebras.