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

Abstract

We introduce the notion of Hilbert $C^{\ast }$-module independence: Let $\mathcal{A}$ be a unital $C^{\ast }$-algebra and let ${\mathcal{E}}_i\subseteq \mathcal{E},i=1,2$, be ternary subspaces of a Hilbert $\mathcal{A}$-module $\mathcal{E}$. Then ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ are said to be Hilbert $C^{\ast }$-module independent if there are positive constants $m$ and $M$ such that for every state ${\phi }_i$ on $⟨{\mathcal{E}}_i,{\mathcal{E}}_i⟩,i=1,2$, there exists a state $\phi$ on $\mathcal{A}$ such that \begin{align*} mφ_i(|x|)\leq φ(|x|) \leq Mφ_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 $⟨{\mathcal{E}}_1,{\mathcal{E}}_1⟩$ has the quasi extension property and $z\in {\mathcal{E}}_1\cap {\mathcal{E}}_2$ with $\Vert z\Vert =1$, 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 $⟨z_0,z_0⟩=1$, then ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ are Hilbert $C^{\ast }$-module independent if and only if $\Vert ⟨x,z_0⟩⟨y,z_0⟩\Vert =\Vert ⟨x,z_0⟩\Vert \Vert ⟨y,z_0⟩\Vert$ for all $x\in {\mathcal{E}}_1$ and $y\in {\mathcal{E}}_2$. We also provide some technical examples and counterexamples to illustrate our results.

23 pages