Let $H$ be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra $A$. It was recently conjectured that there does not exist an equivariant *-homomorphism from $A$ (type-I case) or $H$ (type-II case) to the equivariant noncommutative join C*-algebra $A\circledast^\delta H$. When $A$ is the C*-algebra of functions on a sphere, and $H$ is the C*-algebra of functions on $\mathbb{Z}/2\mathbb{Z}$ acting antipodally on the sphere, then the conjecture of type I becomes the celebrated Borsuk–Ulam theorem. Taking advantage of recent work of Passer, we prove the conjecture of type I for compact quantum groups admitting a non-trivial torsion character. Next, we prove that, if the compact quantum group $(H,\Delta)$ admits a representation whose $K_1$-class is non-trivial and $A$ admits a character, then a stronger version of the type-II conjecture holds: the finitely generated projective module associated with $A\circledast^\delta H$ via this representation is not stably free. In particular, we apply this result to the $q$-deformations of compact connected semisimple Lie groups and to the reduced group C*-algebras of free groups on $n>1$ generators.

中文翻译：

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再谈非交换Borsuk–Ulam型猜想

令$ H $是自由作用于单位C *-代数$ A $的非平凡紧致量子群的C *-代数。最近有人推测，从$ A $（I型情况）或$ H $（II型情况）到等变非交换连接C *-代数$ A \ circledast ^ \ delta不存在等变*同态H $。当$ A $是球面上函数的C *代数，而$ H $是$ \ mathbb {Z} / 2 \ mathbb {Z} $上的函数对角作用于球面上的C *代数时，则I型猜想成为著名的Borsuk–Ulam定理。利用Passer的最新工作，我们证明了I型对于允许非平凡扭转特性的紧凑量子群的猜想。接下来，我们证明，如果紧凑量子组$（H，\ Delta）$接受了$ K_1 $类不平凡而$ A $接受了字符的表示，那么II型猜想的一个更​​强的版本成立：通过此表示与$ A \ circledast ^ \ delta H $相关联的有限生成的投影模块不是稳定自由的。特别是，我们将此结果应用于紧连接的半简单李群的qq形变以及$ n> 1 $生成器上自由群的约简C *代数。