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Isometries between quantum convolution algebras. (English) Zbl 1275.46043
A quantum group is a Hopf-von Neumann algebra $$(M, \Delta)$$ endowed with faithful, normal, semifinite weights $$\varphi, \psi$$ which are ‘left-invariant’ and ‘right-invariant’, respectively. The predual $$M_*$$ is to be thought of as $$L^1(\mathbb{G})$$ and $$M$$ as $$L^{\infty}(\mathbb{G})$$ for the ‘locally compact quantum group’ $$\mathbb{G} = (M, \Delta)$$. An isomorphism between two such quantum groups $$\mathbb{G}_1$$ and $$\mathbb{G}_2$$ is a normal $$*$$-isomorphism between $$L^{\infty}(\mathbb{G}_1)$$ and $$L^{\infty}(\mathbb{G}_2)$$ that intertwines the coproducts. The main result is that, if $$L^1(\mathbb{G}_1)$$ and $$L^1(\mathbb{G}_2)$$ are isometrically isomorphic as algebras, then $$\mathbb{G}_1$$ is isomorphic to either $$\mathbb{G}_2$$ or its commutant $$\mathbb{G}_2'$$, extending the classical results for locally compact groups and also those for Kac algebras. An analogous result is obtained for the ‘quantum measure algebras’ $$M(\mathbb{G}) := C_0(\mathbb{G})^*$$. The results are actually proved in slightly more general settings than the ones given here.

##### MSC:
 46L10 General theory of von Neumann algebras 43A20 $$L^1$$-algebras on groups, semigroups, etc. 43A10 Measure algebras on groups, semigroups, etc. 46L65 Quantizations, deformations for selfadjoint operator algebras 17B37 Quantum groups (quantized enveloping algebras) and related deformations
##### Keywords:
von Neumann algebra; weight; quantum group; intrinsic group; antipode
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