Completely positive multipliers of quantum groups

Abstract

We show that any completely positive multiplier of the convolution algebra of the dual of an operator algebraic quantum group $\G$ (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 $\G$. It follows that there is an order bijection between the completely positive multipliers of $L^1(\G)$ and the positive functionals on the universal quantum group $C_0^u(\G)$. 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${}^{\ast }$-weak${}^{\ast }$-continuous.

18 pages; major rewrite

References in corpus (1)

• Multipliers of locally compact quantum groups via Hilbert C${}^{\ast }$-modules

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• Noncommutative Poisson boundaries and Furstenberg-Hamana boundaries of Drinfeld doubles
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• Convolution semigroups on Rieffel deformations of locally compact quantum groups