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Idempotent States and the Inner Linearity Property

Volume 60 / 2012

Teodor Banica, Uwe Franz, Adam Skalski Bulletin Polish Acad. Sci. Math. 60 (2012), 123-132 MSC: Primary 28C10; Secondary 16W30, 46L65. DOI: 10.4064/ba60-2-3


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.


  • Teodor BanicaDepartment of Mathematics
    Cergy-Pontoise University
    95000 Cergy-Pontoise, France
  • Uwe FranzDepartment of Mathematics
    University of Franche-Comté
    16 route de Gray
    25030 Besançon Cedex, France
  • Adam SkalskiInstitute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    P.O. Box 21
    00-956 Warszawa, Poland

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