Hilbert C∗$C^*$‐module independence

We introduce the notion of Hilbert C∗$C^*$‐module independence: Let A$\mathcal {A}$ be a unital C∗$C^*$‐algebra and let Ei⊆E,i=1,2$\mathcal {E}_i\subseteq \mathcal {E},\,\,i=1, 2$, be ternary subspaces of a Hilbert A$\mathcal {A}$‐module E$\mathcal {E}$. Then, E1$\mathcal {E}_1$ and E2$\mathcal {E}_2$ are said to be Hilbert C∗$C^*$‐module independent if there are positive constants m and M such that for every state φi$\varphi _i$ on ⟨Ei,Ei⟩,i=1,2$\langle \mathcal {E}_i,\mathcal {E}_i\rangle ,\,\,i=1, 2$, there exists a state φ on A$\mathcal {A}$ such that mφi(|x|)≤φ(|x|)≤Mφi(|x|2)12,for allx∈Ei,i=1,2.$$\begin{align*} m\varphi _i(|x|)\le \varphi (|x|) \le M\varphi _i{(|x|^2)}^{\frac{1}{2}},\qquad \mbox{for all\nobreakspace }x\in \mathcal {E}_i, i=1, 2. \end{align*}$$We show that it is a natural generalization of the notion of C∗$C^*$‐independence of C∗$C^*$‐algebras. Moreover, we demonstrate that even in the case of C∗$C^*$‐algebras, this concept of independence is new and has a nice characterization in terms of Hahn–Banach–type extensions. We show that if ⟨E1,E1⟩$\langle \mathcal {E}_1,\mathcal {E}_1\rangle$ has the quasi extension property and z∈E1∩E2$z\in \mathcal {E}_1\cap \mathcal {E}_2$ with ∥z∥=1$\Vert z\Vert =1$, then |z|=1$|z|=1$. Several characterizations of Hilbert C∗$C^*$‐module independence and a new characterization of C∗$C^*$‐independence are given. One of characterizations states that if z0∈E1∩E2$z_0\in \mathcal {E}_1\cap \mathcal {E}_2$ is such that ⟨z0,z0⟩=1$\langle z_0,z_0\rangle =1$, then E1$\mathcal {E}_1$ and E2$\mathcal {E}_2$ are Hilbert C∗$C^*$‐module independent if and only if ∥⟨x,z0⟩⟨y,z0⟩∥=∥⟨x,z0⟩∥∥⟨y,z0⟩∥$\Vert \langle x,z_0\rangle \langle y,z_0\rangle \Vert =\Vert \langle x,z_0\rangle \Vert \,\Vert \langle y,z_0\rangle \Vert$ for all x∈E1$x\in \mathcal {E}_1$ and y∈E2$y\in \mathcal {E}_2$. We also provide some technical examples and counterexamples to illustrate our results.