On the separable quotient problem for Banach spaces

TL;DR

This paper surveys known results related to the open separable quotient problem in Banach spaces, focusing on the existence of specific infinite-dimensional separable quotients in various Banach spaces of functions, operators, and measures.

Contribution

It compiles recent and classical results, including those involving complemented copies of sequence spaces and recent advances by Argyros, Dodos, Kanellopoulos, and Sliwa, providing an updated overview.

Findings

Most results follow from known facts about complemented copies of c_0 and l_p.

Recent results by Argyros, Dodos, Kanellopoulos, and Sliwa are included.

The survey complements a previous 1997 survey by Mujica.

Abstract

While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of a particular infinitedimensional separable quotient in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the results presented are consequence of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c_0 and l_p, for 1 <= p <= \infty. Also recent results of Argyros - Dodos - Kanellopoulos, and Sliwa are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica.