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 $S_{A^+}$ for the intersection of the positive cone and the unit sphere of $A$, we establish that an element $a \in S_{A^+}$ is a projection if, and only if, $$\{a\} = \left\{x\in S_{A^+}: \|x-z\| = 1 \hbox{ for all } z\in S_{A^+}\hbox{ with } \|a-z\|=1 \right\}.$$ 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.