  We introduce the notion of Hilbert $C^*$-module independence: Let
$\mathscr{A}$ be a unital $C^*$-algebra and let $\mathscr{E}_i\subseteq
\mathscr{E},\,\,i=1, 2$, be ternary subspaces of a Hilbert $\mathscr{A}$-module
$\mathscr{E}$. Then $\mathscr{E}_1$ and $\mathscr{E}_2$ are said to be Hilbert
$C^*$-module independent if there are positive constants $m$ and $M$ such that
for every state $\varphi_i$ on $\langle
\mathscr{E}_i,\mathscr{E}_i\rangle,\,\,i=1, 2$, there exists a state $\varphi$
on $\mathscr{A}$ such that \begin{align*} m\varphi_i(|x|)\leq \varphi(|x|) \leq
M\varphi_i(|x|^2)^{\frac{1}{2}},\qquad \mbox{for all~}x\in \mathscr{E}_i, i=1,
2. \end{align*} We show that it is a natural generalization of the notion of
$C^*$-independence of $C^*$-algebras. Moreover, we demonstrate that even in
case of $C^*$-algebras this concept of independence is new and has a nice
characterization in terms of extensions. This enriches the theory of
independence of $C^*$-algebras. We show that if $\langle
\mathscr{E}_1,\mathscr{E}_1\rangle $ has the quasi extension property and $z\in
\mathscr{E}_1\cap \mathscr{E}_2$ with $\|z\|=1$, then $|z|=1$. Several
characterizations of Hilbert $C^*$-module independence and a new
characterization of $C^*$-independence are given. One of characterizations
states that if $z_0\in \mathscr{E}_1\cap \mathscr{E}_2$ is such that $\langle
z_0,z_0\rangle=1$, then $\mathscr{E}_1$ and $\mathscr{E}_2$ are Hilbert
$C^*$-module independent if and only if $\|\langle x,z_0\rangle\langle
y,z_0\rangle\|=\|\langle x,z_0\rangle\|\,\|\langle y,z_0\rangle\|$ for all
$x\in \mathscr{E}_1$ and $y\in \mathscr{E}_2$. We also provide some technical
examples and counterexamples to illustrate our results.
