Examples of compact quantum groups with [formula omitted] a factor

Abstract

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.