Boundary representations, geometry of matrix ranges, and C$^*$-envelopes of finite-dimensional operator systems
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918300802351104 |
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| author | Farenick, Douglas Li, Chi-Kwong Singla, Sushil |
| author_facet | Farenick, Douglas Li, Chi-Kwong Singla, Sushil |
| contents | An analysis of the boundary representations and C$^*$-envelopes of some finite-dimensional operator systems $\mathcal R$ is undertaken by considering relationships between operator-theoretic properties of a $d$-tuple $\mathfrak x=(x_1,\dots, x_d)$ of elements in a unital C$^*$-algebra $\mathcal A$ and operator-system properties of the linear span $\mathcal O_{\mathfrak x}$ of $\{1_\mathcal A,x_1,x_1^*,\dots,x_d,x_d^*\}$. This approach lends itself well to the study of certain phenomena in single- and several-variable operator theory, such as the Smith-Ward property. The matrix range of a $d$-tuple of elements $\mathfrak x$ is matrix-affinely homeomorphic to the matrix state space of the operator system determined by $\mathfrak x$, and many of our methods connect the geometry of these compact matrix convex sets to operator system properties, including various forms of nuclearity and lifting properties. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08322 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary representations, geometry of matrix ranges, and C$^*$-envelopes of finite-dimensional operator systems Farenick, Douglas Li, Chi-Kwong Singla, Sushil Operator Algebras 46L07, 46L05, 47A80, 47L25 An analysis of the boundary representations and C$^*$-envelopes of some finite-dimensional operator systems $\mathcal R$ is undertaken by considering relationships between operator-theoretic properties of a $d$-tuple $\mathfrak x=(x_1,\dots, x_d)$ of elements in a unital C$^*$-algebra $\mathcal A$ and operator-system properties of the linear span $\mathcal O_{\mathfrak x}$ of $\{1_\mathcal A,x_1,x_1^*,\dots,x_d,x_d^*\}$. This approach lends itself well to the study of certain phenomena in single- and several-variable operator theory, such as the Smith-Ward property. The matrix range of a $d$-tuple of elements $\mathfrak x$ is matrix-affinely homeomorphic to the matrix state space of the operator system determined by $\mathfrak x$, and many of our methods connect the geometry of these compact matrix convex sets to operator system properties, including various forms of nuclearity and lifting properties. |
| title | Boundary representations, geometry of matrix ranges, and C$^*$-envelopes of finite-dimensional operator systems |
| topic | Operator Algebras 46L07, 46L05, 47A80, 47L25 |
| url | https://arxiv.org/abs/2510.08322 |