For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism.
Namely, consider the Kadison inequality
$$u(a^* a) \geq u(a)^* u(a),$$
the multiplicative domain $D_u$ of $A$ is given by all the elements $a$ for which this inequality turns into an equality.
For any $d_1, d_2 \in D_u$ and any $ x \in A $, one has
$$ u(d_1x) = u(d_1)u(x),\quad u(xd_2) = u(x) u(d_2),\quad u(d_1xd_2) = u(d_1)u(x)u(d_2). $$
(Cf. Chapter 5.1 of "Tensor Products of $C^*$-Algebras and Operator Spaces" by G. Pisier.)

My question extends to scenarios involving multiple $C^*$-subalgebras, say $B, C \subset A$. I am interested in understanding the conditions under which the CP map $u$ act homomorphically across these subalgebras.
More precisely, what are the necessary and sufficient conditions to ensure 
$$u(b c) = u(b) u(c),$$
for $b \in B$, $c \in C$?
Considerations might include the mutual commutativity of $B$ and $C$, some states on $A$, or some symmetries induced by group actions (w.r.t states or subalgebras), or to von Neumann algebras instead, etc..

I tried some paper and textbook digging but haven't found anything, so any insights, comments or references will be highly appreciated!