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

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.