Distance to regular elements and polar decompositions in a C*-algebra
Hannes Thiel
TL;DR
This work addresses how to quantify the distance from an element $a$ of a C*-algebra $A$ to the set of regular elements $A_{\mathrm{reg}}$ by relating it to polar decompositions of suitable cut-downs. It proves that $\operatorname{dist}(a,A_{\mathrm{reg}})\le\gamma$ is equivalent to the existence, for all $\delta>\gamma$, of a partial isometry $w\in A$ with $w e_\delta = v e_\delta$, where $a=v|a|$ is the canonical polar decomposition in $A^{**}$, and to the polar decomposability in $A$ of both $v f(|a|)$ for continuous $f$ vanishing near $[0,\gamma]$ and the $\delta$-cut-downs $a_\delta$. Building on Moore–Penrose invertibility and 2×2 block-regularity arguments, the paper extends Pedersen’s and Brown–Pedersen’s descriptions of distances to invertible and quasi-invertible elements to the broader setting of regular elements, and strengthens Xue’s approximate-polar-decomposition results by yielding actual polar decompositions for cut-downs. The results provide a robust tool for analyzing the density of regular elements and set the stage for applications to C*-algebras with dense regular elements.
Abstract
We show that the distance from an element of a C*-algebra to the set of regular elements is the infimum of the $δ>0$ for which the $δ$-cut-down of the element admits a polar decomposition within the algebra. This parallels results of Pedersen and Brown-Pedersen describing the distance to invertible and quasi-invertible elements through polar decompositions of cut-downs whose polar parts are unitaries or extreme partial isometries.