A Homological Property and Arens Regularity of Locally Compact Quantum Groups

We characterize two important notions of amenability and compactness of a locally compact quantum group G in terms of certain homological properties. For this, we show that G is character amenable if and only if it is both amenable and co-amenable. We ûnally apply our results to Arens regularity problems of the quantum group algebra L1(G). In particular, we improve an interesting result by Hu, Neufang, and Ruan.