Uniform continuity over locally compact quantum groups.

*(English)*Zbl 1188.46048Let \(G\) be a locally compact group. The C*-algebra \(\text{LUC}(G)\) of left uniformly continuous functions on \(G\) can be written as \(\text{LUC}(G) = L^\infty(G)L^1(G)\), using the natural action of \(L^1(G)\) on its dual. Based on this characterisation, E. E. Granirer [Trans. Am. Math. Soc. 189, 371–382 (1974; Zbl 0292.43015)] introduced the space \(\text{UCB}(\widehat G)\) as the closure of \(A(G)\text{VN}(G)\subseteq \text{VN}(G)\), using the action of the Fourier algebra \(A(G)\) on the left group von Neumann algebra \(\text{VN}(G)\). Similarly, one can define the space of left uniformly continuous elements over a locally compact quantum group. To this end, consider a locally compact quantum group \({\mathbb G} = (M,\Gamma, \phi, \psi)\), where \((M,\Gamma)\) is a Hopf-von Neumann algebra and \(\phi\) and \(\psi\) are left and right Haar weights on \((M,\Gamma)\). The pre-dual \(L^1({\mathbb G})\) of the von Neumann algebra \(L^\infty({\mathbb G}) = M\) is a Banach algebra, which acts naturally on \(L^\infty({\mathbb G})\) in two ways: from the left and from the right. Following the paper under review, define \(\text{LUC}({\mathbb G})\) and \(\text{RUC}({\mathbb G})\) as the closed linear spans of \(L^\infty({\mathbb G})L^1({\mathbb G})\) and \(L^1({\mathbb G})L^\infty({\mathbb G})\), respectively, and define \(\text{UC}({\mathbb G}) = \text{LUC}({\mathbb G})\cap \text{RUC}({\mathbb G})\). (The space \(\text{RUC}({\mathbb G})\) has also appeared in [Z.-G. Hu, M. Neufang and Z.-J. Ruan, “Multipliers on a new class of Banach algebras, locally compact quantum groups, and topological centres”, Proc. Lond. Math. Soc. (3) 100, No. 2, 429–458 (2010; Zbl 1192.43002)].) If \({\mathbb G}\) is given by \(L^\infty(G)\), where \(G\) is an actual locally compact group, then these spaces are just the function spaces consisting of left uniformly continuous functions, right uniformly continuous functions, and uniformly continuous functions. On the dual side, when \({\mathbb G}\) is given by \(\text{VN}(G)\), we obtain Granirer’s \(\text{UCB}(\widehat G)\) (in this case, all three spaces coincide).

In the paper under review, the author studies the spaces \(\text{LUC}({\mathbb G})\), \(\text{RUC}({\mathbb G})\) and \(\text{UC}({\mathbb G})\) and uses these spaces to study the amenability of locally compact quantum groups. The author shows that \(\text{LUC}({\mathbb G})\), \(\text{RUC}({\mathbb G})\) and \(\text{UC}({\mathbb G})\) are always operator systems that contain the reduced C*-algebra \(\text{C}_0({\mathbb G})\) associated with \({\mathbb G}\) and are contained in the multiplier algebra \({\mathcal M}(\text{C}_0({\mathbb G}))\). In both classical cases, given by \(L^\infty(G)\) and \(\text{VN}(G)\), these operator systems are known to be C*-algebras. The author shows that if \({\mathbb G}\) is a co-amenable locally compact quantum group such that \(\text{C}_0({\mathbb G})\) has a bounded approximate identity in its centre, then also \(\text{LUC}({\mathbb G})\), \(\text{RUC}({\mathbb G})\) and \(\text{UC}({\mathbb G})\) are C*-algebras. The space \(\text{WAP}({\mathbb G})\) of weakly almost periodic functionals on \(L^1({\mathbb G})\) is also considered. It is shown that \(\text{WAP}({\mathbb G})\) is an operator system containing \(\text{C}_0({\mathbb G})\), and if \({\mathbb G}\) is co-amenable, \(\text{WAP}({\mathbb G})\) is contained in \(\text{UC}({\mathbb G})\). As a partial answer to a problem posed by E. Bédos and L. Tuset [Int. J. Math. 14, 865–884 (2003; Zbl 1051.46047)], the author proves that a co-amenable \({\mathbb G}\) is amenable if and only if there is a left invariant mean on \({\mathcal M}(\text{C}_0({\mathbb G}))\) if and only if there is a left invariant mean on \(\text{LUC}({\mathbb G})\). The clear writing style makes the paper a pleasure to read.

In the paper under review, the author studies the spaces \(\text{LUC}({\mathbb G})\), \(\text{RUC}({\mathbb G})\) and \(\text{UC}({\mathbb G})\) and uses these spaces to study the amenability of locally compact quantum groups. The author shows that \(\text{LUC}({\mathbb G})\), \(\text{RUC}({\mathbb G})\) and \(\text{UC}({\mathbb G})\) are always operator systems that contain the reduced C*-algebra \(\text{C}_0({\mathbb G})\) associated with \({\mathbb G}\) and are contained in the multiplier algebra \({\mathcal M}(\text{C}_0({\mathbb G}))\). In both classical cases, given by \(L^\infty(G)\) and \(\text{VN}(G)\), these operator systems are known to be C*-algebras. The author shows that if \({\mathbb G}\) is a co-amenable locally compact quantum group such that \(\text{C}_0({\mathbb G})\) has a bounded approximate identity in its centre, then also \(\text{LUC}({\mathbb G})\), \(\text{RUC}({\mathbb G})\) and \(\text{UC}({\mathbb G})\) are C*-algebras. The space \(\text{WAP}({\mathbb G})\) of weakly almost periodic functionals on \(L^1({\mathbb G})\) is also considered. It is shown that \(\text{WAP}({\mathbb G})\) is an operator system containing \(\text{C}_0({\mathbb G})\), and if \({\mathbb G}\) is co-amenable, \(\text{WAP}({\mathbb G})\) is contained in \(\text{UC}({\mathbb G})\). As a partial answer to a problem posed by E. Bédos and L. Tuset [Int. J. Math. 14, 865–884 (2003; Zbl 1051.46047)], the author proves that a co-amenable \({\mathbb G}\) is amenable if and only if there is a left invariant mean on \({\mathcal M}(\text{C}_0({\mathbb G}))\) if and only if there is a left invariant mean on \(\text{LUC}({\mathbb G})\). The clear writing style makes the paper a pleasure to read.

Reviewer: Pekka Salmi (Oulu)

##### MSC:

46L89 | Other “noncommutative” mathematics based on \(C^*\)-algebra theory |

43A07 | Means on groups, semigroups, etc.; amenable groups |

46L07 | Operator spaces and completely bounded maps |

46L65 | Quantizations, deformations for selfadjoint operator algebras |

47L25 | Operator spaces (= matricially normed spaces) |

47L50 | Dual spaces of operator algebras |

81R15 | Operator algebra methods applied to problems in quantum theory |