Lipschitz-free spaces and approximation properties

The main results of this thesis are the following. We prove that the Lipschitzfree space over any closed C1-submanifold of RN has the metric approximation property (MAP), with respect to any norm on RN. We also prove that the Lipschitz-free space over any ‘locally downwards closed’ subset of RN has the MAP, with respect to any norm. Moreover, we show that the Lipschitzfree spaces over two particular subsets of R2, namely a subset containing a cusp at (0, 0), and a subset homeomorphic to the pseudo-arc, have the MAP, with respect to any norm. We also state and prove a useful connection between ‘almost-extension’ operators for Lipschitz functions and linear projection operators, which we use to present some limitations of the known techniques for obtaining the MAP for Lipschitz-free spaces. Another set of results is the following. Let T be a ‘properly metrisable’ topological space, that is, locally compact, separable and metrisable. Let MT be the non-empty set of all proper metrics d on T compatible with its topology, and equipMT with the topology of uniform convergence, where the metrics are regarded as functions on T2. We prove that if T is uncountable then the set ATf of metrics d ∈MT for which the Lipschitz-free space F(T, d) fails the approximation property (AP) is a dense set inMT . Combining this with a result of Dalet, we conclude that for any properly metrisable space T, ATf is either empty or dense in MT . If K = 2N is the standard Cantor space, we also prove that the set of metrics AK,1 for which the Lipschitz-free space F(K, d) has the MAP is a residual Fσδ set in MK. It follows that the set AKf is a dense meager set in MK.