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A representation theorem for locally compact quantum groups. (English) Zbl 1194.22003
Let \(G=(M,\Gamma,\phi,\psi)\) be a locally compact quantum group as defined by J. Kustermans and S. Vaes [Math. Scand. 92, No. 1, 68–92 (2003; Zbl 1034.46067)], where \(M=L_\infty(G)\) is a von Neumann algebra, \(\Gamma\) a co-multiplication and \(\phi\) (resp. \(\psi\)) a left (resp. right) invariant Haar weight. Then one can define the quantum versions of the Banach algebra \(L_1(G)\) and of the Hilbert space \(L_2(G)\). Let \(\hat G\) denote the dual quantum group for \(G\). The authors introduce the algebra \(M_{cb}^r(L_1(G))\) of completely bounded right multipliers on \(L_1(G)\) and show that this algebra can be identified with the algebra of normal completely bounded \(\hat M\)-bimodule maps on the space of bounded operators on \(L_2(G)\) which leave \(M\) invariant. This is used to show that quantum group duality can be expressed purely as a commutation relation.

22D15 Group algebras of locally compact groups
22D20 Representations of group algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
22D35 Duality theorems for locally compact groups
43A10 Measure algebras on groups, semigroups, etc.
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
46L07 Operator spaces and completely bounded maps
46L10 General theory of von Neumann algebras
46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
47L10 Algebras of operators on Banach spaces and other topological linear spaces
47L25 Operator spaces (= matricially normed spaces)
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