$L_p+L_\infty$ and $L_p\cap L_\infty$ are not isomorphic for all $1\le p<\infty,$ $p\ne 2$

Abstract

Isomorphic classification of symmetric spaces is an important problem related to the study of symmetric structures in arbitrary Banach spaces. This research was initiated in the seminal work of Johnson, Maurey, Schechtman and Tzafriri (JMST, 1979). Somewhat later it was extended by Kalton to lattice structures (1993). In particular, in JMST (see also Lindenstrauss-Tzafriri book [1979, Section 2.f]) it was shown that the space $L_2 \cap L_p$ for $2 \leq p < \infty$ (resp. $L_2+L_p$ for $1 < p \leq 2$) is isomorphic to $L_p$. A detailed investigation of various properties of separable sums and intersections of $L_p$-spaces (i.e., with $p<\infty$) was undertaken by Dilworth in the papers from 1988 and 1990. In contrast to that, we focus here on the problem if the nonseparable spaces $L_p +L_{\infty}$ and $L_p \cap L_{\infty}$, $1\le p<\infty$, are isomorphic or not. We prove that these spaces are not isomorphic if $1 \leq p < \infty$, $p \neq 2$. It comes as a consequence of the fact that the space $L_p \cap L_{\infty}$, $1\le p<\infty$, $p\ne 2$, does not contain a complemented subspace isomorphic to $L_p$. In particular, as a subproduct, we show that $L_p \cap L_{\infty}$ contains a complemented subspace isomorphic to $l_2$ if and only if $p = 2$. The problem if $L_2 +L_{\infty}$ and $L_2 \cap L_{\infty}$ are isomorphic or not remains open.