Linear extensions and linear liftings in subspaces of $C(X)$

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If X is a compact Hausdorff space, if B is a closed subspace of C(X) and if F is a closed subset of X, conditions are given which ensure the existence of a linear extension operator of norm 1 from the restriction space BIF to B. 1. In this note we study the following two problems. Given a compact Hausdorff space X and a uniformly closed subspace B of C(X), the space of all continuous complex-valued functions on X, which separates points of X and contains the constant functions. I. If F is a closed subset of X with certain properties defined below (F is an M-set), when is there a linear extension operator of norm 1 from the restriction space BI F into B? II. If J is a subspace of B which is an M-ideal, does there exist a continuous linear inverse to the canonical map of B into B/J? M-sets which are considered in the first problem are the complex-space analogues of split faces of compact convex sets and they are also the function space analogues of the generalized peak sets for function algebras. An M-ideal in B is just the space of functions which vanish on some M-set for B so that the second problem is another formulation of the first once it has been seen that B F is isometrically isomorphic to B/J. In this note we show that for an M-set F problem I has a solution if B F has the metric approximation property. We then use a construction of Davie [8] to show that if F is metric and if problem I has a solution in general then B F has the metric approximation property. The idea in the proof of the existence of the linear extension operator is to show that one can construct finite dimensional extensions of the same kind as the ones constructed in the papers of Michael and Pelczyn'ski [10] and Davie [8]. The construction of the finite dimensional extensions involves among others ideas from a paper of Rao [11] and the construction is actually reduced to showing the existence of an extension of a single function on a new M-set in a space larger than X, the proof of the existence of this extension is then a slight modification of the proofs in the papers of Alfsen and Hirsberg [3] and the author and Rao [7]. In the case when BIF is a a1-space, solutions to problems I and II can be Received by the editors June 19, 1974. AMS (MOS) subject classifications (1970). Primary 46E15; Secondary 46A30.