“Homological manifestations of quantum group amenability”
We present a non-relative homological characterization of quantum group amenability in terms of 1-injectivity of the dual L∞(Gˆ) as an operator module over its predual L1(Gˆ). This result not only establishes the equivalence between amenability of a locally compact group G and 1-injectivity of its group von Neumann algebra L(G) as an A(G)-module, it provides a novel tool for the development of abstract harmonic analysis on locally compact quantum groups. Indeed, we present several applications, including a proof that closed quantum subgroups (in the sense of Vaes) of amenable quantum groups are amenable; a decomposability result for completely bounded L1(Gˆ)-module maps on L∞(Gˆ), and a simplified proof that amenable discrete quantum groups have co-amenable compact duals which avoids the use of modular theory. Time permitting, we will also discuss recent work on relative biflatness of the Fourier algebra.