Journal of Operator Theory

Volume 45, Issue 2, Spring 2001 pp. 357-376.

Weak $C^*$-Hopf algebras and multiplicative isometries

Authors: Gabriella Bohm (1), and Kornel Szlachanyi (2)
Author institution: (1) Research Institute for Particle and Nuclear Physics, P.O.B. 49, H-1525 Budapest 114, Hungary
(2) Research Institute for Particle and Nuclear Physics, P.O.B. 49, H-1525 Budapest 114, Hungary

Summary: We show how the data of a finite dimensional weak $C^*$-Hopf algebra can be encoded into a pair $(\mathcal H,V)$ where $\mathcal H$ is a finite dimensional Hilbert space and $V\colon\mathcal H\otimes\mathcal H\to\mathcal H\otimes\mathcal H$ is a partial isometry satisfying, among others, the pentagon equation. In case of $V$ being unitary we recover the Baaj-Skandalis multiplicative unitary of the discrete compact type. Relation with the pseudomultiplicative unitary approach proposed by J.-M. Vallin and M. Enock is also discussed.

Keywords: Multiplicative partial isometries, pseudo-multiplicative unitaries, weak Hopf algebras

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