Geometric Interpolation in n-Tuples of Noncommutative $$L_p$$ -Spaces

Let \(\mathcal {M}\) be a von Neumann algebra with a normal faithful semifinite trace. In this paper, we consider that in n-tuples of noncommutative \(L_p\)-spaces \(l_s^{(n)}(L_p(\mathcal {M}))\), the norm is invariant under the action of invertible elements in \(\mathcal {M}\). Then we prove that the complex interpolating theorem in the case of \(l_s^{(n)}(L_p(\mathcal {M}))\). Using this result, we obtain that Clarkson’s inequalities for n-tuples of operators with weighted norm of noncommutative \(L_p\)-spaces, where the weight being a positive invertible operator in \(\mathcal {M}\).