Examples of compact quantum groups with L ∞ ( G ) a factor

Krajczok, J. and Sołtan, P. M. (2024) Examples of compact quantum groups with L ∞ ( G ) a factor. Journal of Functional Analysis, 286(6), 110297. (doi: 10.1016/j.jfa.2023.110297)

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For each λ∈] 0,1] we exhibit an uncountable family of compact quantum groups G such that the von Neumann algebra L∞ (G) is the injective factor of type IIIλ with separable predual. We also show that uncountably many injective factors of type III0 arise as L∞ (G) for some compact quantum group G. We introduce natural invariants of quantum groups related to the scaling group, modeled on the Connes invariant T for von Neumann algebras. We compute the values of these invariants in our examples, which allows us to distinguish between them. We also investigate their connection to the Connes invariants T(L∞(G)), S(L∞(G)). In the final section we show that for a compact quantum group G the von Neumann algebra L∞(G) cannot have a direct summand of the form B(H) with dim H=∞. In particular, factors of type I (of any dimension) cannot be obtained as L∞(G) for a non-trivial compact quantum group G.

Item Type:Articles
Additional Information:Research presented in this paper was partially supported by the Polish National Agency for Academic Exchange, Polonium grant PPN/BIL/2018/1/00197 as well as by the FWO–PAS project VS02619N: von Neumann algebras arising from quantum symmetries. Additionally the first author was supported by EPSRC grants EP/T03064X/1 and EP/T030992/1 and the second author by NCN (National Science Centre, Poland) grant no. 2022/47/B/ST1/00582.
Glasgow Author(s) Enlighten ID:Krajczok, Dr Jacek
Authors: Krajczok, J., and Sołtan, P. M.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Functional Analysis
ISSN (Online):1096-0783
Published Online:05 January 2024

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Project CodeAward NoProject NamePrincipal InvestigatorFunder's NameFunder RefLead Dept
309470Quantum groups in actionChristian VoigtEngineering and Physical Sciences Research Council (EPSRC)EP/T03064X/1M&S - Mathematics