Uncountable almost irredundant sets in nonseparable C*-algebras

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.