Banach spaces with many boundedly complete basic sequences failing PCP

TL;DR

This paper constructs Banach spaces that do not contain , fail the point of continuity property, yet every semi-normalized basic sequence has a boundedly complete subsequence, challenging previous assumptions.

Contribution

It provides a counterexample of Banach spaces with specific properties, answering an open problem posed by Rosenthal.

Findings

Existence of Banach spaces without  containing the specified properties.

Counterexample to the assumption that PCP implies certain sequence properties.

Demonstrates that boundedly complete subsequences can exist without  presence.

Abstract

We prove that there exist Banach spaces not containing $\ell_1$, failing the point of continuity property and satisfying that every semi-normalized basic sequence has a boundedly complete basic subsequence. This answers in the negative the problem of the Remark 2 in H. P. Rosenthal. "Boundedly complete weak-Cauchy sequences in Banach spaces with PCP." J. Funct. Anal. 253 (2007) 772-781.