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We show that the analyticity of semigroups $(T_t)_{t \geq 0}$ of (notnecessarily positive) selfadjoint contractive Fourier multipliers on$\mathrm{L}^p$-spaces of any abelian locally compact group is preserved by thetensorisation of the identity operator $\mathrm{Id}_X$ of a Banach space $X$for a large class of K-convex Banach spaces, answering partially a conjectureof Pisier. The result is even new for semigroups of Fourier multipliers actingon $\mathrm{L}^p(\mathbb{R}^n)$. The proof relies on the use of noncommutativeBanach spaces and we give a more general result for semigroups of Fouriermultipliers acting on noncommutative $\mathrm{L}^p$-spaces. Finally, we alsogive a somewhat different version of this result in the discrete case, i.e. forRitt operators.

On analyticity of semigroups on Bochner spaces and on vector-valued noncommutative $\mathrm{L}^p$-spaces