AI 中文总结
研究通过将哈里斯论证中相关部分替换，把四维算子系统换成三维超刚性算子系统，得到反例说明三维算子系统不具有史密斯 - 沃德性质，此系统还非精确，是首个三维非精确算子系统的例子。
AI 中文摘要
哈里斯表明到卡尔金代数的不可提升的单射表示给出四维算子系统反例。本文隔离相关论证部分，用三维超刚性算子系统替换四维的，所得卡尔金子系统无单位完全正提升，即给出史密斯 - 沃德问题反例，且此系统非精确。
英文摘要
Harris recently showed that a non-liftable injective representation into the Calkin algebra gives explicit four-dimensional operator systems in the Calkin algebra without the lifting property, and hence a counterexamples to the generalized Smith--Ward problem for four-dimensional operator systems. The main obstruction also appears in an earlier work by Paulsen on this problem. We isolate the relevant part of this argument and replace the four-dimensional operator system by a three-dimensional hyperrigid operator system inside a matrix amplification of \[ C_r^*(\F_2). \] The resulting Calkin subsystem is of the form span$\{1,q(D),q(K)\}$, where $D$ and $K$ are selfadjoint operators, and the identity map on this operator system has no unital completely positive lift. Equivalently, the operator $D+iK$ gives a counterexample to the Smith--Ward problem. By a result of Kavruk, the dual of this operator system fails to be exact, and hence is the first example of a three-dimensional operator system that is not exact.
Comments9 pages; replaced S in Corollary 4.3 with its dual; added keywords and fundings
