Von Neumann algebraic Quantum Object is direct sum of type I factors

Question

I am looking at the non-standard quantum projective spaces $A:=\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. Now I want to show that if I take the von Neumann algebra completion of this, denoted by $\mathscr{L}^{\infty}(A)$, is isomorphic to the direct sum of type I factors, in other words: $$\mathscr{L}^{\infty}(A)\cong\bigoplus_{i\in I}B(\mathcal{H}_i)$$ for some index set $I$ and some Hilbert spaces $\mathcal{H}_i$ labeled by $I$. I think one can use Theorem $3.20$ from "Representation Theory of Quantized Enveloping Algebras with Interpolating Real Structure" by I don't know how to show the ingredients. Maybe there are also other ideas to prove that. Thank you very much.