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Summary:
Summary: We show that the symmetrization of a brace algebra structure yields the structure of a symmetric brace algebra. We also show that the symmetrization of the natural brace structure on $\bigoplus_{k\ge 1}\operatorname{Hom}(V^{\otimes k},V)$ coincides with the natural symmetric brace structure on $\bigoplus_{k\ge 1}\operatorname{Hom}(V^{\otimes k},V)^{as}$, the direct sum of spaces of antisymmetric maps $V^{\otimes k}\to V$.
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