On two long standing open problems on Lp-spaces

Keywords:
Lp-spaces, complemented subspace, unconditional basisAbstract
The present note was written during the preparation of the talk at the International Conference dedicated to 70-th anniversary of Professor O. Lopushansky, September 16-19, 2019, Ivano-Frankivsk (Ukraine). We focus on two long standing open problems. The first one, due to Lindenstrauss and Rosenthal (1969), asks of whether every complemented infinite dimensional subspace of L1 is isomorphic to either L1 or ℓ1. The second problem was posed by Enflo and Rosenthal in 1973: does there exist a nonseparable space Lp(μ) with finite atomless μ and 1<p<∞, p≠2, having an unconditional basis? We analyze partial results and discuss on some natural ideas to solve these problems.