Necessity of condition (*) for simplicity of fixed-point algebras in quasi-free actions on O_n

Determine whether condition (*)—that for every irreducible unitary representation π of the compact group G there exists n ≥ 0 such that π is contained in the tensor power ρ^⊗n of a faithful unitary representation ρ: G → U(n)—is necessary for the fixed-point algebra O_n^α of the quasi-free action α of G on the Cuntz algebra O_n to be purely infinite and simple.

Background

In Section 7 the paper studies quasi-free actions of a compact group G on the Cuntz algebras O_n and O_∞. For O_n, given a faithful unitary representation ρ: G → U(n), the associated quasi-free action α on O_n is considered, and the authors introduce condition (*): for every irreducible unitary representation π of G, π appears in some tensor power ρ^⊗n.

The paper explains that when ρ(G) ⊂ SU(n), existing results already imply simplicity of the fixed-point algebra O_n^α. Using a graph-algebra approach, the authors show that condition (*) suffices to ensure that O_n^α is purely infinite and simple, which in turn (via their main theorems) implies α is isometrically shift-absorbing and quasi-product.

However, the authors point out that while condition (*) is sufficient, it is not established whether it is necessary for O_n^α to be purely infinite and simple, and they highlight this as an explicit unknown.