Extensions of Operator-Valued Kernels on $\mathbb{F}^{+}_{d}$

Abstract: We study the problem of extending a positive-definite operator-valued kernel, defined on words of a fixed finite length from a free semigroup, to a global kernel defined on all words. We show that if the initial kernel satisfies a natural one-step dominance inequality on its interior, a global extension that preserves this interior data and the dominance property is always possible. This extension is constructed explicitly via a Cuntz-Toeplitz model. For the problem of matching the kernel on the boundary, we introduce an intrinsic shift-consistency condition. We prove this condition is sufficient to guarantee the existence of a global extension that agrees with the original kernel on its entire domain.