Some metric and homotopy properties of partial isometries

TL;DR

This paper investigates metric inequalities and homotopy properties of partial isometries in C*-algebras, establishing sharp bounds and conditions for homotopy and continuity of associated maps.

Contribution

It introduces new sharp inequalities for partial isometries and characterizes homotopy conditions based on norm distances, extending understanding of their geometric and topological properties.

Findings

Established that ||u*u - v*v|| ; ||u - v|| for partial isometries

Proved homotopy between partial isometries when ||u - v|| < 1 or 2 for extremal cases

Analyzed continuity points of the partial isometry map in C*-algebras

Abstract

We show that ||u*u - v*v|| \leq ||u - v|| for partial isometries u and v. There is a stronger inequality if both u and v are extreme points of the unit ball of a C*-algebra, and both inequalities are sharp. If u and v are partial isometries in a C*-algebra A such that ||u - v|| < 1, then u and v are homotopic through partial isometries in A. If both u and v are extremal, then it is sufficient that ||u - v|| < 2. The constants 1 and 2 are both sharp. We also discuss the continuity points of the map which assigns to each closed range element of A the partial isometry in its canonical polar decomposition.