Unconditional and bimonotone structures in high density Banach spaces

TL;DR

This paper explores the existence of long unconditional and bimonotone basic sequences in high density Banach spaces using transfinite topological games and stabilization techniques, revealing conditions for their existence.

Contribution

It introduces transfinite topological game analysis and stabilization methods to establish the presence of unconditional and bimonotone sequences in high density Banach spaces.

Findings

Existence of long unconditional basic sequences in high density Banach spaces.

Conditions under which uncountable basic sequences admit bimonotone subsequences.

Application of stabilization of projectional resolutions to achieve regularity properties.

Abstract

It is shown that every normalized weakly null sequence of length $\kappa_{\lambda}$ in a Banach space has a subsequence of length $\lambda$ which is an unconditional basic sequence; here $\kappa_{\lambda}$ is a large cardinal depending on a given infinite cardinal $\lambda$. Transfinite topological games on Banach spaces are analyzed which determine the existence of a long unconditional basic sequence. Then 'asymptotic disentanglement' condition in a transfinite setting is studied which ensures a winning strategy for the unconditional basic sequence builder in the above game. The following problem is investigated: When does a Markushevich basic sequence with length uncountable regular cardinal $\kappa$ admit a subsequence of the same length which is a bimonotone basic sequence? Stabilizations of projectional resolutions of the identity (PRI) are performed under a density contravariance principle to gain some additional strong regularity properties, such as bimonotonicity.