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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!

Edit: I should mention the motivation. I consider the case when $A$ is the space of bounded operators of some Hilbert space $\mathcal{H} = \mathcal{H}_B \otimes \mathcal{H}_C$, with $B$ and $C$ the bounded operators of $\mathcal{H}_B$ and $\mathcal{H}_C$ respectively. Let $\mathcal{H}' = \mathcal{H}_B' \otimes \mathcal{H}_C'$, and $V_X: \mathcal{H}_X \to \mathcal{H}_X'$ isometries for $X = B, C$. Then the map $u(b \otimes c) = (V_B b V_B^*) \otimes (V_C c V_C^*)$ is an example.

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!

Edit: I should mention the motivation. I consider the case when $A$ is the space of bounded operators of some Hilbert space $\mathcal{H} = \mathcal{H}_B \otimes \mathcal{H}_C$, with $B$ and $C$ the bounded operators of $\mathcal{H}_B$ and $\mathcal{H}_C$ respectively. Let $\mathcal{H}' = \mathcal{H}_B' \otimes \mathcal{H}_C'$, and $V_X: \mathcal{H}_X' \to \mathcal{H}_X$ isometries for $X = B, C$. Then the map $u(b \otimes c) = (V_B^* b V_B) \otimes (V_C^* c V_C)$ is an example.

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!

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!

Edit: I should mention the motivation. I consider the case when $A$ is the space of bounded operators of some Hilbert space $\mathcal{H} = \mathcal{H}_B \otimes \mathcal{H}_C$, with $B$ and $C$ the bounded operators of $\mathcal{H}_B$ and $\mathcal{H}_C$ respectively. Let $\mathcal{H}' = \mathcal{H}_B' \otimes \mathcal{H}_C'$, and $V_X: \mathcal{H}_X \to \mathcal{H}_X'$ isometries for $X = B, C$. Then the map $u(b \otimes c) = (V_B b V_B^*) \otimes (V_C c V_C^*)$ is an example.