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Noncommutative Borsuk–Ulam-type conjectures revisited
Journal of Noncommutative Geometry ( IF 0.7 ) Pub Date : 2021-01-18 , DOI: 10.4171/jncg/352
Ludwik Dąbrowski 1 , Piotr Hajac 2 , Sergey Neshveyev 3
Affiliation  

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:

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。
更新日期:2021-02-08
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