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Non-degenerate *-homorphism from $A$ (or $M(A)$) to $M(B)$ are strict (where non-degenerate means that $\phi(A)B$ is total in $B$). An important property of strict maps $\phi:A\to M(B)$ is that they …

Locally compact quantum groups can be used to construct quantum channels, see the recent preprint http://arxiv.org/abs/1210.2738 by Jason Crann and Matthias Neufang.

Is there a reason why the power in the equation and the size of the matrices are the same? E.g., if you take two anti-commuting roots of $I_n$, you will always have $(xA+yB)^2 = (x^2+y^2)I_n$. More g …

How about $\lambda_k = \frac{U'_k(n)}{U_k(n)}$ where the $U_k$ denote the Chebyshev polynomials of the second kind, $U_0(x)=1$, $U_1(x)=x$, and $U_k(x)=xU_{k-1}(x)-U_{k-2}(x)$ for $k\ge 2$. In Sect …

A little side remark, concerning the last part of your question: Marc Rosso describes in Marc Ross, Algèbres enveloppantes quantifiées, groupes quantiques compacts de matrices et calcul différentiel …