Extensions of Operator-Valued Kernels on $\mathbb{F}^{+}_{d}$
James Tian
TL;DR
This work addresses extending a positive-definite operator-valued kernel $K$ defined on words of bounded length in the free semigroup to a global kernel on $\mathbb{F}^+_d$, under the natural one-step dominance $K_\Sigma\le K$. It develops a constructive dilation framework: from truncated data on $\Lambda_N$ it builds a Kolmogorov RKHS, identifies an interior density, and dilates the interior row-contraction to a row isometry to obtain a global kernel with a Cuntz-Toeplitz realization $\widetilde{K}(\alpha,\beta)=W^* S^\alpha P (S^\beta)^* W$. A shift-consistency condition on the level-$N$ boundary is shown to be sufficient for full boundary agreement, enabling $\widetilde{K}(\alpha,\beta)=K(\alpha,\beta)$ on $\Lambda_N$. The results connect with noncommutative moment problems and classical Hausdorff moments (for $d=1$) and are illustrated by finite-dimensional examples that demonstrate both the sufficiency and limitations of the boundary condition. Overall, the paper provides a dilation-theoretic, constructive approach to noncommutative moment/interpolation problems with explicit realizations and clear boundary-matching criteria.
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.