A remark on the behaviour ofL p-multipliers and the range of operators acting onL p-spaces

In this paper, new results are obtained concerning the uniform approximation property (UAP) inL ^p-spaces (p≠2,1,∞). First, it is shown that the “uniform approximation function” does not allow a polynomial estimate. This fact is rather surprising since it disproves the analogy between UAP-features and the presence of “large” euclidian subspaces in the space and its dual. The examples are translation invariant spaces on the Cantor group and this extra structure permits one to replace the problem with statements about the nonexistence of certain multipliers in harmonic analysis. Secondly, it is proved that the UAP-function has an exponential upper estimate (this was known forp=1, ∞). The argument uses Schauder’s fix point theorem. Its precise behaviour is left unclarified here. It appears as a difficult question, even in the translation invariant context.