On two long standing open problems on $L_p$-spaces

M.M. Popov

doi:10.15330/cmp.12.1.229-241

M.M. Popov Vasyl Stefanyk Precarpathian National University, 57 Shevchenka str., 76018, Ivano-Frankivsk, Ukraine, Pomeranian University in Słupsk, 76-200, Słupsk, Poland https://orcid.org/0000-0002-3165-5822

https://doi.org/10.15330/cmp.12.1.229-241

Keywords: $L_p$-spaces, complemented subspace, unconditional basis

Published online: 2020-06-29

Abstract

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 $L_1$ is isomorphic to either $L_1$ or $\ell_1$. The second problem was posed by Enflo and Rosenthal in 1973: does there exist a nonseparable space $L_p(\mu)$ with finite atomless $\mu$ and $1<p<\infty$, $p \neq 2$, having an unconditional basis? We analyze partial results and discuss on some natural ideas to solve these problems.