# Article

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Keywords:
loop; quasigroup; sphere; Hilbert space; spherical geometry
Summary:
On the unit sphere \$\Bbb S\$ in a real Hilbert space \$\bold H\$, we derive a binary operation \$\odot \$ such that \$(\Bbb S,\odot )\$ is a power-associative Kikkawa left loop with two-sided identity \$\bold 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 \$\Bbb S\$. \$(\Bbb S,\odot )\$ is not a loop, and the right translations which fail to be injective are easily characterized. \$(\Bbb 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 \$-\bold e_{0}\$ where they have a nonremovable discontinuity. The orthogonal group \$O(\bold H)\$ is a semidirect product of \$(\Bbb S,\odot )\$ with its automorphism group. The left loop structure of \$(\Bbb S,\odot )\$ gives some insight into spherical geometry.
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