Approximation by invertible elements and the generalized $E$-stable rank for $A({\boldsymbol D})_{\mathsf R}$ and $C({\boldsymbol D})_{\mathrm{sym}}$

Abstract

We determine the generalized $E$-stable ranks for the real algebra, $C(\boldsymbol{D})_{\mathrm{sym}}$, of all complex valued continuous functions on the closed unit disk, symmetric to the real axis, and its subalgebra $A(\boldsymbol{D})_{\mathsf R}$ of holomorphic functions. A characterization of those invertible functions in $C(E)$ is given that can be uniformly approximated on $E$ by invertibles in $A(\boldsymbol {D})_{\mathsf R}$. Finally, we compute the Bass and topological stable rank of $C(K)_{\mathrm{sym}}$ for real symmetric compact planar sets $K$.