Distance to regular elements and polar decompositions in a C*-algebra

TL;DR

This paper characterizes the distance from an element in a C*-algebra to the set of regular elements using polar decompositions of cut-downs, extending previous results on invertible and quasi-invertible elements.

Contribution

It establishes a new characterization of the distance to regular elements via polar decompositions, generalizing earlier work on invertible elements in C*-algebras.

Findings

Distance to regular elements equals the infimum of δ for which the δ-cut-down admits a polar decomposition.

Extends Pedersen and Brown-Pedersen's results on invertible elements to regular elements.

Provides a new geometric perspective on the structure of C*-algebras.

Abstract

We show that the distance from an element of a C*-algebra to the set of regular elements is the infimum of the $\delta>0$ for which the $\delta$-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.