Uncountable almost irredundant sets in nonseparable C*-algebras

TL;DR

This paper demonstrates that under PFA, every nonseparable scattered C*-algebra contains an uncountable almost irredundant set, revealing new structural properties of such algebras.

Contribution

It establishes the existence of uncountable almost irredundant sets in nonseparable C*-algebras with certain ideal structures under PFA, a novel result in the field.

Findings

Uncountable almost irredundant sets exist in certain nonseparable C*-algebras.

Under PFA, every nonseparable scattered C*-algebra admits an uncountable almost irredundant set.

The size of almost irredundant sets is bounded above by the density of the algebra.

Abstract

In this article, we consider the notion of almost irredundant sets: A subset $\mathcal{X}$ of a C*-algebra $\mathcal{A}$ is called almost irredundant if and only if for every $a\in \mathcal{X}$, the element $a$ does not belong to the norm-closure of $$\{\sum_{i=1}^n \lambda_i \prod_{j=1}^{n_i}a_{{i,j}}: \textrm{ where } a_{{i,j}} \in \mathcal{X}\setminus\{a\} \textrm{ and} \sum |\lambda_i|\leq 1\}.$$ Since every almost irrredundant set is in particular a discrete set, it follows that the density of $\mathcal{A}$ is an upper bound for the size of almost irredundant sets. We prove that under the Proper Forcing Axiom (PFA), there is an uncountable almost irredundant set in every C*-algebra with an uncountable increasing sequence of ideals. In particular, assuming PFA, every nonseparable scattered C*-algebra admits an uncountable almost irredundant set.