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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.

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
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