  We consider the space of odd spinors on the circle, and a decomposition into
spinors supported on either the top or on the bottom half of the circle. If an
operator preserves this decomposition, and acts on the bottom half in the same
way as a second operator acts on the top half, then the fusion of both
operators is a third operator acting on the top half like the first, and on the
bottom half like the second. Fusion restricts to the Banach-Lie group of
restricted orthogonal operators, which supports a central extension of
implementers on a Fock space. In this article, we construct a lift of fusion to
this central extension. Our construction uses Tomita-Takesaki theory for the
Clifford-von Neumann algebras of the decomposed space of spinors. Our
motivation is to obtain an operator-algebraic model for the basic central
extension of the loop group of the spin group, on which the fusion of
implementers induces a fusion product in the sense considered in the context of
transgression and string geometry. In upcoming work we will use this model to
construct a fusion product on a spinor bundle on the loop space of a string
manifold, completing a construction proposed by Stolz and Teichner.
