6 Schur product
Given quantum sets \((B_1,m_1,\eta _1,\sigma _r)\) and \((B_2,m_2,\eta _2,\vartheta _r)\), we define the Schur product operator \(\cdot \bullet \cdot \colon \mathcal{B}(\mathcal{B}(B_1,B_2),\mathcal{B}(\mathcal{B}(B_1,B_2),\mathcal{B}(B_1,B_2)))\) as the map \(m_2(x\otimes y)m_1^*\).
Given \(a,c\in (B_1,m_1,\eta _1,\sigma _r)\) and \(b,d\in (B_2,m_2,\eta _2,\vartheta _r)\), we get
\[ \left\lvert a\right\rangle \! \! \left\langle b\right\rvert \bullet \left\lvert c\right\rangle \! \! \left\langle d\right\rvert =\left\lvert ac\right\rangle \! \! \left\langle bd\right\rvert . \]
Proof
\begin{align*} {\left\lvert a\right\rangle \! \! \left\langle b\right\rvert }\bullet {\left\lvert c\right\rangle \! \! \left\langle d\right\rvert } & = m(\left\lvert a\right\rangle \! \! \left\langle b\right\rvert \otimes \left\lvert c\right\rangle \! \! \left\langle d\right\rvert )m^* = m\left\lvert a\otimes c\right\rangle \! \! \left\langle b\otimes d\right\rvert m^*\\ & = \left\lvert m(a\otimes c)\right\rangle \! \! \left\langle m(b\otimes d)\right\rvert =\left\lvert ac\right\rangle \! \! \left\langle bd\right\rvert . \end{align*}
Given linear maps \(A_1,A_2,A_3\colon X_1\to X_2\), we get
\[ \left({A_1}\bullet {A_2}\right)\bullet {A_3}={A_1}\bullet \left({A_2}\bullet {A_3}\right). \]
Proof
This follows from the associativity and co-associativity properties of the multiplication maps.
For any \(r,t\in \mathbb {R}\) and linear maps \(f\) and \(g\) from quantum set \(A\) to quantum set \(B\), we get
\[ \Psi _{r,t}({f}\bullet {g})=\Psi _{r,t}(f)\Psi _{r,t}(g). \]
Proof
Straightforward computation.