Amenable actions of locally compact groups

Abstract

We consider a locally compact group G with jointly continuous action G × Z → Z on a locally compact space. The finite Radon (regular Borel) measures M( G) act naturally on various function spaces supported on Z, including the continuous bounded functions CB( Z). If Z supports a (not necessarily unique) nonnegative quasi-invariant measure ν we apply some recent studies [ 7] to define a Banach module action of M( G) on L 1( Z, ν), and this induces a natural adjoint action on L ∞(Z, ν) = L 1(Z, ν) ∗ . These actions of M( G) (and of G) give us definitions of amenability of the action of G on various function spaces supported on Z, including CB( Z) and L ∞( Z, ν) (when there is a quasi-invariant ν on Z), corresponding to the existence of a left-invariant mean on these various spaces. When G acts on one of its coset spaces solG H , H a closed subgroup, there is always a quasi-invariant measure on G H and the different definitions of amenability of the action G × G H → G H all coincide. A number of manipulations of invariant means on groups (the case where G = Z) then carry over to the context of transformation groups. We apply them to explore the following problems. Let G × G H → G H be an amenable action of G on one of its coset spaces. Let ν be a quasi-invariant measure on G H .