A Metric Characterization of Projections Among Positive Norm-One Elements in Unital C^*-Algebras

We characterize projections among positive norm-one elements in unital C^* C ∗ -algebras in purely geometric terms determined by the norm of the underlying Banach space. Concretely, let A be a C^* C ∗ -algebra (or a JB^* JB ∗ -algebra) whose positive cone and unit sphere are denoted by A⁺ A + and SA S A , respectively. The positive portion of the unit sphere in A , denoted by S_A⁺ S A + , is the set A⁺ ∩ SA A + ∩ S A , while the unit sphere of positive norm-one elements around a subset S S in SA⁺ S A + is the set aligned Sph__S_A⁺ (S) := \ x∈ S_A⁺ : x-s =1 for all s∈ S \. aligned Sph S A + ( S ) : = { x ∈ S A + : ‖ x - s ‖ = 1 for all s ∈ S <m