Idempotent States and the Inner Linearity Property
Volume 60 / 2012
Bulletin Polish Acad. Sci. Math. 60 (2012), 123-132
MSC: Primary 28C10; Secondary 16W30, 46L65.
DOI: 10.4064/ba60-2-3
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 Cesàro limit of the linear functional $\varphi={\rm tr}\circ\pi$. We then discuss some consequences of this result, notably to inner linearity questions.
