@MISC{Kinyon00(1.1), author = {Michael K. Kinyon}, title = {(1.1)}, year = {2000} }

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Abstract

Abstract. 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.