We prove that the L 2 -Betti numbers of a unimodular locally compact group G coincide, up to a natural scaling constant, with the L 2 -Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of G. As a consequence, we obtain that the reduced and un-reduced L 2 -Betti numbers of G agree and that the L 2 -Betti numbers of a lattice Γ in G equal those of G up to scaling by the covolume of Γ in G. We also deduce several vanishing results, including the vanishing of the reduced L 2 -cohomology for amenable locally compact groups.

Utilizing the notion of property (T) we construct new examples of quantum group norms on the polynomial algebra of a compact quantum group, and provide criteria ensuring that these are not equal to neither the minimal nor the maximal norm. Along the way we generalize several classical operator algebraic characterizations of property (T) to the quantum group setting which unify recent approaches to property (T) for quantum groups with previous ones. The techniques developed furthermore provide tools to answer two open problems; firstly a question by Bédos, Murphy and Tuset about automatic continuity of the comultiplication and secondly a problem left open by Woronowicz regarding the structure of elements whose coproduct is a finite sum of simple tensors. Date

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A notion of L 2 -homology for compact quantum groups is introduced, generalizing the classical notion for countable, discrete groups. If the compact quantum group in question has tracial Haar state, it is possible to define its L 2 -Betti numbers and Novikov-Shubin invariants/capacities. It is proved that these L 2 -Betti numbers vanish for the Gelfand dual of a compact Lie group and that the zeroth Novikov-Shubin invariant equals the dimension of the underlying Lie group. Finally, we relate our approach to the approach of A. Connes and D. Shlyakhtenko by proving that the L 2 -Betti numbers of a compact quantum group, with tracial Haar state, are equal to the Connes-Shlyakhtenko L 2 -Betti numbers of its Hopf * -algebra of matrix coefficients.