I remember that I had asked a question about obstructions for a $n^2$ dimensional vector bundle for which the fibers admit a structure of matrix algebra. I do not remember whether it was an independent question or it was included in comments in another question.

I searched a lot both in web and in Mo to find that question but I can not find it. I remember a participant says that there are obstructions as Diximier Duady classes in 3th cohohomology of the base space.

I would like to read that conversation as a supplementary to the following my MO question

Parallel transport as algebra isomorphism

Can one help my to find that conversation?

Thank you very much

Diximierin those comments? (You have also used various spellings of Douady in your question and your comments here on meta.) $\endgroup$