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

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 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 a 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.