Fund. Math. 194 (2007), no. 1, 15–44. For a thorough outline of the problematics of liftings of category algebras (definitions, history of results and links to previous papers), see the review [M. R. Burke et al., Topology Appl. 153 (2006), no. 7, 1164–1191; MR2203028 (2007f:54050)], supplemented by [M. R. Burke, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1075–1082; MR1128726 (93e:28006)]. The main result of the present article shows that if a non-discrete separable metric topological group G is weakly a-favourable, then the category algebra of G has no left-invariant Borel lifting. It is further shown that under the continuum hypothesis, many groups in the above class admit dense Baire subgroups which have left-invariant Borel lifting. On the other hand, it is consistent to assume the existence of a separable metric Baire group whose category algebra group does not admit a Borel lifting
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