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We prove the norm estimates for operator-valued functions on free groups supported on the words with fixed length ($f={\sum }_{\left|w\right|=l}{a}_w\otimes \lambda \left(w\right)$). Next, we replace the translations by the free generators with a free family of operators and prove inequalities of the same type.

For $\left(P_k\right)$ being Rademacher, Fermion or q-Gaussian (-1 ≤ q ≤ 0) operators, we find the optimal constants $C_{2n}$, n∈ ℕ, in the inequality $\parallel {\sum }_{k=1}^N{A}_k\otimes P_k{\parallel }_{2n}\le {\left[C_{2n}\right]}^{1/2n}max{\parallel ({\sum }_{k=1}^{N}A{*}_k{A}_k}^{1/2}{\parallel }_{L_{2n}},\parallel ({\sum }_{k=1}^N{A}_{k}A{*}_k$1/2∥L2n$,$valid for all finite sequences of operators $\left(A_k\right)$ in the non-commutative $L_{2n}$ space related to a semifinite von Neumann algebra with trace. In particular, $C_{2n}=(2nr-1)!!$ for the Rademacher and Fermion sequences.