Four-dimensional operator systems without the lifting property

Mathematics > Operator Algebras [Submitted on 31 Jul 2025 (v1), last revised 15 Jun 2026 (this version, v2)] Title:Four-dimensional operator systems without the lifting property View PDF HTML (experimental)Abstract:The purpose of this note is to provide a family of explicit examples of $4$-dimensional operator systems contained in the Calkin algebra $\mathcal{Q}(\mathcal{H})$ on a separable infinite-dimensional Hilbert space $\mathcal{H}$ for which the identity map has no unital completely positive (ucp) lift to $\mathcal{B}(\mathcal{H})$ with respect to the canonical quotient map $\pi:\mathcal{B}(\mathcal{H}) \to \mathcal{Q}(\mathcal{H})$. More specifically, to each unital $C^*$-algebra $\mathcal{A}$ generated by $n$ unitaries and unital $*$-homomorphism $\rho:\mathcal{A} \to \mathcal{Q}(\mathcal{H})$ with no ucp lift, we construct a four-dimensional operator subsystem $\mathcal{S}$ of $M_{n+1}(\mathcal{A})$ without the lifting property. As a result, for each $n \geq 2$ we exhibit a four-dimensional operator system $\mathcal{S}$ in $M_{n+1}(C_r^*(\mathbb{F}_n))$ without the lifting property. We also obtain explicit examples where the generalized Smith-Ward problem for liftings of joint matrix ranges for three self-adjoint operators has a negative answer. Submission history From: Samuel Harris [view email][v1] Thu, 31 Jul 2025 19:06:43 UTC (11 KB) [v2] Mon, 15 Jun 2026 18:56:29 UTC (12 KB) Current browse context: math.OA References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.