Moment kernels, nested defects, and Cuntz dilations
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915669017100288 |
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| author | Tian, James |
| author_facet | Tian, James |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_06309 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moment kernels, nested defects, and Cuntz dilations Tian, James Functional Analysis Operator Algebras Primary: 47A20, secondary: 47A13, 47A45, 47A60 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. |
| title | Moment kernels, nested defects, and Cuntz dilations |
| topic | Functional Analysis Operator Algebras Primary: 47A20, secondary: 47A13, 47A45, 47A60 |
| url | https://arxiv.org/abs/2509.06309 |