Completely positive multipliers of quantum groups. (English) Zbl 1282.43002
Summary: We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group \(\mathbb G\) (either a locally compact quantum group, or a quantum group coming from a modular or manageable multiplicative unitary) is induced in a canonical fashion by a unitary corepresentation of \(\mathbb G\). It follows that there is an order bijection between the completely positive multipliers of \(L^1(\mathbb G)\) and the positive functionals on the universal quantum group \(C^u_0(\mathbb G)\). We provide a direct link between the Junge, Neufang, Ruan representation result and the representing element of a multiplier, and use this to show that their representation map is always weak*-weak*-continuous.
MSC:
| 43A22 | Homomorphisms and multipliers of function spaces on groups, semigroups, etc. |
| 46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |
| 22D15 | Group algebras of locally compact groups |
| 22D25 | \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations |
| 81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |
Keywords:
locally compact quantum group; completely bounded multiplier; completely positive multiplier; corepresentationReferences:
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