Ideal Submodules versus Ternary Ideals versus Linking Ideals

TL;DR

This paper establishes the equivalence of ideal submodules and ternary ideals in Hilbert modules, explores their properties, and introduces ternary extensions, providing new insights and open problems in the structure theory of Hilbert modules.

Contribution

It demonstrates the equivalence of ideal submodules and ternary ideals in Hilbert modules and develops a framework for ternary ideals and extensions.

Findings

Ideal submodules and ternary ideals are the same in Hilbert modules.

Ternary ideals are suitable for defining modules' ideals without referencing the algebra.

Introduction of ternary extensions with foundational properties and open problems.

Abstract

We show that ideal submodules and closed ternary ideals in Hilbert modules are the same. We use this insight as a little peg on which to hang a little note about interrelations with other notions regarding Hilbert modules. In Section 3, we show that the ternary ideals (and equivalent notions) merit fully, in terms of homomorphisms and quotients, to be called ideals of (not necessarily full) Hilbert modules. The properties to be checked are intrinsically formulated for the modules (without any reference to the algebra over which they are modules) in terms of their ternary structure. The proofs, instead, are motivated from a third equivalent notion, linking ideals (Section 2), and a Theorem (Section 3) that all extends nicely to (reduced) linking algebras. As an application, in Section 4, we introduce ternary extensions of Hilbert modules and prove most of the basic properties (some new even for the known notion of extensions of Hilbert modules), by reducing their proof to the well-known analogue theorems about extensions of $C^*$-algebras. Finally, in Section 5, we propose several open problems that our method naturally suggests.