On the non-uniqueness of minimal projection in spaces

The main objective of this note is to exhibit a simple example of subspaces UL_p(μ) (p≠2) that admit two different projections with minimal norm. While for p=1,∞, such subspaces are well-known W. Odyniec, G. Lewicki, Minimal Projections in Banach Spaces, in: Lecture Notes in Mathematics, vol. 1449, Springer-Verlag, Berlin, 1990. Problems of existence and uniqueness and their application], for 1<p<∞ their existence was open.