The norm reveals projections in the positive unit sphere of an arbitrary C*-algebra

We show how a simple metric condition in terms of the C^*-norm of an arbitrary C^*-algebra A can be employed to determine those norm-one positive elements which are projections. Concretely, if we write SA⁺ for the intersection of the positive cone and the unit sphere of A, we establish that an element a ∈ SA⁺ is a projection if, and only if, \ = ∈ SA⁺: \|x-z\| = 1 for all z∈ SA⁺ with \|a-z\|=1 \. This proves a conjecture which has been previously confirmed for von Neumann algebras and unital C^*-algebras. In the setting of not necessarily unital JB^*-algebras, the Shirshov--Cohn theorem is invoked to obtain a similar metric characterisation of projections. In the wider setting of JB^*-triples, we show how a metric characterisation in terms of inner ideals can be established to determine non-zero tripotents.