Characterizing projections among positive operators in the unit sphere

Abstract

Let E and P be subsets of a Banach space X, and let us define the unit sphere around E in P as the set

Sph(E; P) := {x is an element of P : parallel to x - b parallel to = 1 for all b is an element of E}.

Given a C*-algebra A and a subset E subset of A; we shall write Sph(+) (E) or Sph(A)(+) (E) for the set Sph(E; S(A(+))); where S(A(+)) denotes the unit sphere of A(+). We prove that, for every complex Hilbert space H, the following statements are equivalent for every positive element a in the unit sphere of B(H):

(a) a is a projection;

(b) Sph(B)((H))(+) (Sph(B)((H))(+) ({a})) = {a}.

We also prove that the equivalence remains true when B(H) is replaced with an atomic von Neumann algebra or with K(H-2), where H-2 is an infinite-dimensional and separable complex Hilbert space.