Global left loop structures on spheres

On the unit sphere S in a real Hilbert space H, we derive a binary operation ⊙ such that (S,⊙) is a power-associative Kikkawa left loop with two-sided identity e0, i.e., it has the left inverse, automorphic inverse, and Al properties. The operation ⊙ is compatible with the symmetric space structure of S. (S,⊙) is not a loop, and the right translations which fail to be injective are easily characterized. (S,⊙) satisfies the left power alternative and left Bol identities “almost everywhere” but not everywhere. Left translations are everywhere analytic; right translations are analytic except at −e0 where they have a nonremovable discontinuity. The orthogonal group O(H) is a semidirect product of (S,⊙) with its automorphism group. The left loop structure of (S,⊙) gives some insight into spherical geometry.