A Counterexample To Universal Character Density For Compact Quantum Groups

Abstract

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.