locally compact quantum group

Definition 0.1.

A locally compact quantum group defined as in ref. [1] is a quadruple $\mathrm{G}\mathrm{=}\mathrm{(}A\mathrm{,}\mathrm{\Delta }\mathrm{,}\mu \mathrm{,}\nu \mathrm{)}$, where $A$ is either a $C^{*}$– or a$W^{*}$– algebra equipped with a co-associative comultiplication (http://planetmath.org/WeakHopfCAlgebra2)$\mathrm{\Delta }\mathrm{:}A\mathrm{\to }A\mathrm{\otimes }A$ and two faithful semi-finite normal weights, $\mu$ and $\nu$ –right and -left Haar measures.

0.0.1 Examples

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

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