Lower and upper local uniform $K$-monotonicity in symmetric spaces

TL;DR

This paper explores the structure of symmetric spaces through local and global $K$-monotonicity properties, establishing connections with order continuity and the Kadec-Klee property, and clarifying conditions for various forms of local uniform $K$-monotonicity.

Contribution

It provides new insights into the relationships between local and global $K$-monotonicity, order continuity, and the Kadec-Klee property in symmetric spaces, including conditions for monotonicity implications.

Findings

Established a relationship between strict $K$-monotonicity and local uniform $K$-monotonicity.

Identified conditions under which upper local uniform $K$-monotonicity implies upper local uniform monotonicity.

Demonstrated correlations between $K$-order continuity and lower local uniform $K$-monotonicity.

Abstract

Using the local approach to the global structure of a symmetric space $E$ we establish a relationship between strict $K$- monotonicity, lower (resp. upper) local uniform $K$-monotonicity, order continuity and the Kadec-Klee property for global convergence in measure. We also answer the question under which condition upper local uniform $K$-monotonicity concludes upper local uniform monotonicity. Finally, we present a correlation between $K$-order continuity and lower local uniform $K$-monotonicity in a symmetric space $E$ under some additional assumptions on $E$.