Moment kernels, nested defects, and Cuntz dilations

Abstract

Random operator tuples possess a rich second-moment structure that is not visible at the level of pointwise operator inequalities. This paper shows that their averaged word moments form a positive kernel whose behavior is controlled by a single shift-positivity condition. When this condition holds, the kernel admits a Cuntz dilation, and all mean-square interactions are realized inside a canonical isometric model. This leads to a mean-square version of the free von Neumann inequality and to a free functional calculus for random tuples. We further introduce a hierarchy of higher order defects of the moment kernel and prove that their positivity is equivalent to the existence of a nested chain of projections inside one Cuntz dilation. This yields a multi-level decomposition of moment structure, a Wold-type splitting into dissipative and unitary parts, and a curvature-type invariant that measures the asymptotic non-dissipating content of the tuple.