关于超刚性和非退化C*-对应的关系
On hyperrigidity and non-degenerate C*-correspondences
AI总结：
本文通过证明Katsura理想非退化等价于张量代数超刚性，回答了关于非退化C*-对应超刚性的核心问题，并简化了相关证明。
摘要：
我们重新审视Kim以及Katsoulis和Ramsey关于非退化C*-对应超刚性的结果。我们证明张量代数是超刚性的，当且仅当Katsura的理想作用非退化，当且仅当Katsura的理想在任何表示下都非退化。这回答了Katsoulis和Ramsey的问题，证明了他们关于张量代数超刚性必要条件和充分条件是等价的。Kim还证明了Katsura理想左作用的非退化等价于与C*-对应相关的自共轭算子空间的超刚性，我们的方法也提供了这一结果的简化证明。在此过程中，我们研究了Werner意义下的自共轭算子空间的单位化，并回顾了Arveson将极大性与唯一扩展性质和超刚性联系起来的准则，结合Salomon关于生成集的工作。
英文摘要：
We revisit the results of Kim, and of Katsoulis and Ramsey concerning hyperrigidity for non-degenerate C*-correspondences. We show that the tensor algebra is hyperrigid, if and only if Katsura's ideal acts non-degenerately, if and only if Katsura's ideal acts non-degenerately under any representation. This gives a positive answer to the question of Katsoulis and Ramsey, showing that their necessary condition and their sufficient condition for hyperrigidity of the tensor algebra are equivalent. Non-degeneracy of the left action of Katsura's ideal was also shown by Kim to be equivalent to hyperrigidity for the selfadjoint operator space associated with the C*-correspondence, and our approach provides a simplified proof of this result as well. In the process we study unitisations of selfadjoint operator spaces in the sense of Werner, and revisit Arveson's criterion connecting maximality with the unique extension property and hyperrigidity, in conjunction with the work of Salomon on generating sets.
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