Uncountable almost irredundant sets in nonseparable C*-algebras

摘要

In this article, we consider the notion of almost irredundant sets: A subset X of a C*-algebra A is called almost irredundant if and only if for every a is an element of X, the element a does not belong to the norm-closure of{Sigma(n)(i=1) lambda(i) Pi(ni)(j=1) a(i,j) : where a(i,j) is an element of X {a} and Sigma vertical bar lambda(i)vertical bar = 1}.Since every almost irredundant set is in particular a discrete set, it follows that the density of 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. (C) 2021 Elsevier B.V. All rights reserved.