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 $$\{\sum _{i=1}^n{\lambda }_i\prod _{j=1}^{n_i}{a}_{i,j}:\text{ where }{a}_{i,j}\in \mathcal{X}\setminus \{a\}\text{ and}\sum \mathrm{\mid }{\lambda }_i\mathrm{\mid }\le 1\}\mathrm{.}$$ 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.