  On the unit sphere $\mathbb{S}$ in a real Hilbert space $\mathbf{H}$, we
derive a binary operation $\odot$ such that $(\mathbb{S},\odot)$ is a
power-associative Kikkawa left loop with two-sided identity $\mathbf{e}_0$,
i.e., it has the left inverse, automorphic inverse, and $A_l$ properties. The
operation $\odot$ is compatible with the symmetric space structure of
$\mathbb{S}$. $(\mathbb{S},\odot)$ is not a loop, and the right translations
which fail to be injective are easily characterized. $(\mathbb{S},\odot)$
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 $-\mathbf{e}_0$ where they have a
nonremovable discontinuity. The orthogonal group $O(\mathbf{H})$ is a
semidirect product of $(\mathbb{S},\odot)$ with its automorphism group (cf.
http://www.arxiv.org/abs/math.GR/9907085). The left loop structure of
$(\mathbb{S},\odot)$ gives some insight into spherical geometry.
