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Fields of locally compact quantum groups: continuity and pushouts. (English) Zbl 1482.46087

The present article studies continuous fields of \(\mathrm{C}^*\)-algebras arising from closed central (quantum) subgroups, generalising a classical result for amenable locally compact groups from [J. A. Packer and I. Raeburn, Trans. Am. Math. Soc. 334, No. 2, 685–718 (1992; Zbl 0786.22009)]. As a first result, Theorem 0.1 states that if \(G\) is a locally compact quantum subgroup with coamenable dual and \(H \leq G\) is a closed central quantum subgroup such that \(G/H\) is coamenable, then the quantum group \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*_{\mathrm{red}}(G) \cong \mathrm{C}^{\mathrm{r}}_0(\hat G)\) is a continuous \(\mathrm{C}^{\mathrm{r}}_0(\hat H)\)-algebra. This result is applied to prove Theorem 0.2, which says that certain amalgamated free product \(\mathrm{C}^*\)-algebras constructed from discrete quantum groups also give rise to a continuous field of \(\mathrm{C}^*\)-algebras. Interestingly, the author discusses in Section 3 a possible gap in a previous article on continuous fields of \(\mathrm{C}^*\)-algebras arising from amalgamated free products [E. Blanchard, Proc. Edinb. Math. Soc., II. Ser. 52, No. 1, 23–36 (2009; Zbl 1168.46034)], and offers an alternative proof for a result needed from this reference.

MSC:

46L67 Quantum groups (operator algebraic aspects)
46L09 Free products of \(C^*\)-algebras
20G42 Quantum groups (quantized function algebras) and their representations
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
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References:

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