Idempotent states and the inner linearity property

TL;DR

This paper provides an analytic formulation of the Hopf image using idempotent states, linking it to the convolution Cesàro limit of a trace-based functional, with implications for inner linearity in Hopf $C^*$-algebras.

Contribution

It introduces a new analytic approach to understanding the Hopf image via idempotent states and explores consequences for inner linearity properties.

Findings

Idempotent state associated to Hopf image equals convolution Cesàro limit of trace functional

Results connect Hopf image structure with inner linearity questions

Provides a new perspective on the analytic formulation of Hopf images

Abstract

We find an analytic formulation of the notion of Hopf image, in terms of the associated idempotent state. More precisely, if $\pi:A\to M_n(\mathbb C)$ is a finite dimensional representation of a Hopf $C^*$-algebra, we prove that the idempotent state associated to its Hopf image $A'$ must be the convolution Ces\`aro limit of the linear functional $\phi=tr\circ\pi$. We discuss then some consequences of this result, notably to inner linearity questions.