Hilbert C∗$C^*$‐module independence

Abstract

AbstractWe introduce the notion of Hilbert ‐module independence: Let be a unital ‐algebra and let , be ternary subspaces of a Hilbert ‐module . Then, and are said to be Hilbert ‐module independent if there are positive constants m and M such that for every state on , there exists a state φ on such that We show that it is a natural generalization of the notion of ‐independence of ‐algebras. Moreover, we demonstrate that even in the case of ‐algebras, this concept of independence is new and has a nice characterization in terms of Hahn–Banach–type extensions. We show that if has the quasi extension property and with , then . Several characterizations of Hilbert ‐module independence and a new characterization of ‐independence are given. One of characterizations states that if is such that , then and are Hilbert ‐module independent if and only if for all and . We also provide some technical examples and counterexamples to illustrate our results.