  We show that the analyticity of semigroups $(T_t)_{t \geq 0}$ of (not
necessarily positive) selfadjoint contractive Fourier multipliers on
$\mathrm{L}^p$-spaces of any abelian locally compact group is preserved by the
tensorisation of the identity operator $\mathrm{Id}_X$ of a Banach space $X$
for a large class of K-convex Banach spaces, answering partially a conjecture
of Pisier. The result is even new for semigroups of Fourier multipliers acting
on $\mathrm{L}^p(\mathbb{R}^n)$. The proof relies on the use of noncommutative
Banach spaces and we give a more general result for semigroups of Fourier
multipliers acting on noncommutative $\mathrm{L}^p$-spaces. Finally, we also
give a somewhat different version of this result in the discrete case, i.e. for
Ritt operators.

Functional Analysis