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

For each lambda is an element of ]0, 1] we exhibit an uncountable family of compact quantum groups G such that the von Neumann algebra L degrees degrees(G) is the injective factor of type III lambda with separable predual. We also show that uncountably many injective factors of type III0 arise as L degrees degrees(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 degrees degrees(G)), S(L degrees degrees(G)). In the final section we show that for a compact quantum group G the von Neumann algebra L degrees degrees(G) cannot have a direct summand of the form B(H) with dim H = infinity. In particular, factors of type I (of any dimension) cannot be obtained as L degrees degrees(G) for a non-trivial compact quantum group G. (c) 2024 Elsevier Inc. All rights reserved.