  In this paper, we investigate the nuclear trace of vector-valued Fourier
multipliers on the torus and its applications to the index theory of periodic
pseudo-differential operators. First, we characterise the nuclearity of
pseudo-differential operators acting on Bochner integrable functions. In this
regards, we consider the periodic and the discrete cases. We go on to address
the problem of finding sharp sufficient conditions for the nuclearity of
vector-valued Fourier multipliers on the torus. We end our investigation with
two index formulae. First, we express the index of a vector-valued Fourier
multiplier in terms of its operator-valued symbol and then we use this formula
for expressing the index of certain elliptic operators belonging to periodic
H\"ormander classes.
