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

$(\Hil,V)$ where

$\Hil$ is a finite dimensional Hilbert space and

$V\colon\Hil\o\Hil\to\Hil\o\Hil$ 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.

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