2016/03/22 by Brown, Lawrence G.
#46L05 #47B99 #47C15 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1603.07002
We show that ||u*u - v*v|| ≤ ||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.