Hilbert $C^{\ast }$-module independence

We introduce the notion of Hilbert $C^{\ast }$-module independence: Let $\mathcal{A}$ be a unital $C^{\ast }$-algebra and let $\mathscr{E}_i\subseteq \mathscr{E},,,i=1, 2$,beternarysubspacesofaHilbert$\mathscr{A}$-module$\mathscr{E}$\mathrm{.}Then$\mathscr{E}_1$and$\mathscr{E}_2$aresaidtobeHilbert$C^$-moduleindependentiftherearepositiveconstants$m$and$M$ such that for every state ${\phi }_i$ on $\langle \mathscr{E}_i,\mathscr{E}_i\rangle,,,i=1, 2$,thereexistsastate$\varphi$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 $C^{\ast }$-independence of $C^{\ast }$-algebras. Moreover, we demonstrate that even in case of $C^{\ast }$-algebras this concept of independence is new and has a nice characterization in terms of extensions. This enriches the theory of independence of $C^{\ast }$-algebras. We show that if $\langle \mathscr{E}_1,\mathscr{E}_1\rangle $hasthequasiextensionpropertyand$z\in \mathscr{E}_1\cap \mathscr{E}_2$with$|z|=1$,then$|z|=1$. Several characterizations of Hilbert $C^{\ast }$-module independence and a new characterization of $C^{\ast }$-independence are given. One of characterizations states that if $z_0\in {\mathcal{E}}_1\cap {\mathcal{E}}_2$ is such that $\langle z_0,z_0\rangle=1$,then$\mathscr{E}_1$and$\mathscr{E}_2$areHilbert$C^*$-moduleindependentifandonlyif$|\langle x,z_0\rangle\langle y,z_0\rangle|=|\langle x,z_0\rangle|,|\langle y,z_0\rangle|$forall$x\in \mathscr{E}_1$and$y\in \mathscr{E}_2$. We also provide some technical examples and counterexamples to illustrate our results.