2022/05/10 by Benjamin Anderson-Sackaney, Anderson-Sackaney, Benjamin
Mathematics · Physics and Astronomy · #43A10 (Secondary) #46L55 #46L67 (Primary) 46L30 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA) #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2205.05176
openalex publication_date 2022/05/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
We investigate the tracial states and \mathbbG-invariant states on the reduced C^*-algebra Cr(\widehat\mathbbG) of a discrete quantum group \mathbbG. Here, we denote its dual compact quantum group by \widehat\mathbbG. Our main result is that a state on Cr(\widehat\mathbbG) is tracial if and only if it is \mathbbG-invariant. This generalizes a known fact for unimodular discrete quantum groups and builds upon the work of Kalantar, Kasprzak, Skalski, and Vergnioux. As one consequence of this, we find that Cr(\widehat\mathbbG) is nuclear and admits a tracial state if and only if \mathbbG is amenable. This resolves an open problem due to C.-K. Ng and Viselter, and Crann, in the discrete case. As another consequence, we prove that tracial states on Cr(\widehat\mathbbG) "concentrate" on \widehat\mathbbGF, where \mathbbGF is the cokernel of the Furstenberg boundary. Furthermore, given certain assumptions, we characterize the existence of traces on Cr(\widehat\mathbbG) in terms of whether or not \widehat\mathbbGF is Kac type. We also characterize the uniqueness of (idempotent) traces in terms of whether not \widehat\mathbbGF is equal to the canonical Kac quotient of \widehat\mathbbG. These results rely on the following, of which we give proofs: Sołtan's canonical Kac quotient construction, whether it is applied to the universal or the reduced CQG C^*-algebra of \widehat\mathbbG (when the latter admits a trace), yields the maximal Kac type closed quantum subgroup of \widehat\mathbbG.