Some Structural Properties of Homomorphism Dilation Systems for Linear Maps

TL;DR

This paper develops an algebraic framework for dilation theory of linear systems on unital algebras, introducing canonical and universal dilations, and classifies minimal dilations via subspace analysis.

Contribution

It introduces a pure algebraic approach to dilation theory, defining canonical and universal dilations, and classifies minimal dilations through kernel subspaces.

Findings

Every linearly minimal dilation is equivalent to a reduced homomorphism dilation of the universal dilation.

All minimal dilations are classified by subspaces in the kernel of the universal dilation's synthesis operator.

The approach generalizes projection valued dilations to a broader algebraic setting.

Abstract

Inspired by some recent development on the theory about projection valued dilations for operator valued measures or more generally bounded homomorphism dilations for bounded linear maps on Banach algebras, we explore a pure algebraic version of the dilation theory for linear systems acting on unital algebras and vector spaces. By introducing two natural dilation structures, namely the canonical and the universal dilation systems, we prove that every linearly minimal dilation is equivalent to a reduced homomorphism dilation of the universal dilation, and all the linearly minimal homomorphism dilations can be classified by the associated reduced subspaces contained in the kernel of synthesis operator for the universal dilation.