AI 中文总结
该研究针对Woronowicz提出的紧量子群通用特征密度问题给出否定答案，构造出Kac型紧矩阵量子群的反例，明确了通用与约化特征理论的边界。
AI 中文摘要
1987年，Woronowicz提出问题：紧量子群的环境C*代数的余交换部分中，不可约特征的线性张成是否在该部分中范数稠密。经过39年，我们对该问题的通用形式给出了否定答案，即使对于Kac型紧矩阵量子群也是如此。我们从一个具有性质T的无限有限生成单群和一个有限双特征扭出发，构造了一个紧量子群，其通用C*代数是一个全群C*代数。我们证明，Kazhdan投影在扭余积下仍保持余交换，而由诱导表示产生的一个矢量态可将该投影与每个代数余交换元素分离开。定量而言，该Kazhdan投影到不可约特征的闭线性张成的距离至少为1/2。此外，该投影属于约化态射的核。因此，这种阻碍是真正通用的，且在过渡到约化紧量子群后消失，此时已知的特征密度定理仍然成立。该构造确定了通用特征理论与约化特征理论之间的清晰边界，表明仅Kac对称性无法控制通用完备化中的余交换元素。
英文摘要
In 1987, Woronowicz asked whether the linear span of irreducible characters is norm dense in the cocommutative part of the ambient C-star algebra of a compact quantum group. After thirty-nine calendar years, we give a negative answer to the universal form of this question, even for compact matrix quantum groups of Kac type. Starting from an infinite finitely generated simple group with property T and a finite bicharacter twist, we construct a compact quantum group whose universal C-star algebra is a full group C-star algebra. A Kazhdan projection is shown to remain cocommutative under the twisted coproduct, while a vector state arising from an induced representation separates this projection from every algebraic cocommutative element. Quantitatively, the distance from the Kazhdan projection to the closed linear span of irreducible characters is at least one half. Moreover, the projection lies in the kernel of the reducing morphism. The obstruction is therefore genuinely universal and disappears after passage to the reduced compact quantum group, where the known character-density theorem remains valid. The construction identifies a sharp boundary between universal and reduced character theory and shows that Kac symmetry alone does not control cocommutative elements in the universal completion.
