  We consider the time-harmonic electromagnetic transmission problem for the
unit sphere. Appealing to a vector spherical harmonics analysis, we prove the
first stability result of the local multiple trace formulation (MTF) for
electromagnetics, originally introduced by Hiptmair and Jerez-Hanckes [Adv.
Comp. Math. 37 (2012), 37-91] for the acoustic case, paving the way towards an
extension to general piecewise homogeneous scatterers. Moreover, we investigate
preconditioning techniques for the local MTF scheme and study the accumulation
points of induced operators. In particular, we propose a novel second-order
inverse approximation of the operator. Numerical experiments validate our
claims and confirm the relevance of the preconditioning strategies.
