Computing norms of free operators with matrix coefficients

Let g₁, g₂, . . . , g_n be the generators of the free group F_n and denote by λ its left regular representation. A formula for norms of operators of the form Σ a_i ⊗ λ(g_i) where a_i are complex matrices or operators is derived similarly to the formula of Akemann and Ostrand for the scalar case. This formula can be used to compute norms of arbitrary finitely supported convolution operators on the free group. Essentially the same technique yields a formula for the norms of sums of matrix valued creation and annihilation operators on full Fock space. Finally some results about the numerical evaluation of the formula are proved.