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\(C^*\)-algebraic quantum groups arising from algebraic quantum groups. (English) Zbl 1009.46038

From the introduction: The main purpose of this paper is the construction of a \(C^*\)-algebraic quantum group (in the sense of Masuda, Nakagami and Woronowicz (unpublished)) out of a multiplier Hopf \(*\)-algebra which possesses a positive left-invariant functional. In the first section we give an overview of the results of van Daele about such multiplier Hopf \(*\)-algebras. In the second section we introduce the \(C^*\)-algebra together with the comultiplication. From there on, we gradually prove that this \(C^*\)-algebra fits almost in the scheme of Masuda, Nakagami and Woronowicz.
Multiplier Hopf algebras have been introduced by A. Van Daele [Trans. Am. Math. Soc. 342, 917-932 (1994; Zbl 0809.16047)] as a generalisation of Hopf algebras to the case of nonunital algebras.

MSC:

46L89 Other “noncommutative” mathematics based on \(C^*\)-algebra theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
46L05 General theory of \(C^*\)-algebras
16W30 Hopf algebras (associative rings and algebras) (MSC2000)

Citations:

Zbl 0809.16047
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