locally compact quantum group

Definition 0.1 A locally compact quantum group defined as in ref. [1] is a quadruple $ QCG_l =(A, \Delta, \mu, \nu)$, where $ A$ is either a $ C^*$- or a $ W^*$ - algebra equipped with a co-associative comultiplication $ \Delta: A \to A \otimes A$ and two faithful semi-finite normal weights, $ \mu$ and $ \nu$ - right and -left Haar measures.

Examples

  1. An ordinary unimodular group $ G$ with Haar measure $ \mu$: $ A = L^{\infty}(G, \mu), \Delta: f(g) \mapsto f(gh)$, $ S: f(g) \mapsto f(g^{}-1), \phi(f) = \int_G f(g)d\mu (g)$, where $ g, h \in G, f \in L^{\infty} (G, \mu)$.

  2. A = Ł(G) is the von Neumann algebra generated by left-translations $ L_g$ or by left convolutions $ L_f = \int_G (f(g)L_g d \mu (g))$ with continuous functions $ f(\dot) \in L^1(G,\mu) \Delta: \mapsto L_g \otimes L_g \mapsto L_g^{-1}, \phi(f) = f(e) $, where $ g \in G$, and $ e$ is the unit of $ G$.

Bibliography

1
Leonid Vainerman. 2003.Locally Compact Quantum Groups and Groupoids: Proceedings of the Meeting of Theoretical Physicists and Mathematicians, Strasbourg, February 21-23, 2002 Series in Mathematics and Theoretical Physics, 2, Series ed. V. Turaev., Walter de Gruyter Gmbh et Co: Berlin.



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