My question will be about the non-standard quantum projective space $\mathcal{A}_q(\mathbb{CP}^n(c,d))$ introduced by Dijkhuizen and Noumi. I want to see this algebra now on a von Neumann algebraic level. In particular, I will study the space $\mathscr{L}^{\infty}(\mathcal{A}_q(\mathbb{CP}^n(c,d)))$. From my point of view, this von Neumann algebra has to be isomorphic to a direct sum of type $I$ factors, in particular:
$$\mathscr{L}^{\infty}(\mathcal{A}_q(\mathbb{CP}^n(c,d)))\cong\bigoplus_{m=1}^p{B(\mathcal{H}_i})$$
for a certain finite $p\in\mathbb{N}$ and certain Hilbert spaces $\mathcal{H}_i$ (I think they has to the $\ell^2(\mathbb{N}^r)$ for certain $r\in\mathbb{N}$ (by searching for explicit representations)). The isomorphism is (from my point of view) a statement which follows by Theorem $3.20$ of "http://arxiv.org/pdf/1307.3642.pdf". But can one give also an explicit isomorphism? How to realize $\mathscr{L}^{\infty}(\mathcal{A}_q(\mathbb{CP}^n(c,d)))$ as von Neumann algebra?
Thanks a lot.
