On covering number of C*-algebras

We introduce two new invariants for C*-algebras: the approximate covering number, ACov(A, m), and the unit approximate covering number, UACov(A, m). Additionally, we define three tracial covering numbers: the tracial covering number, the tracial approximate covering number, and the tracial unit approximate covering number. We establish equivalent conditions for a number to be the approximate covering number or the unit approximate covering number. For unital Z-stable C*-algebras and simple, unital, separable AF-algebras, covering numbers are shown to be at most 2. Furthermore, we demonstrate that if a C*-algebra A is weakly purely infinite and ACov(A, m) = 1 for all m ≥ 2, then A is purely infinite. If the approximate covering number or unit approximate covering number of A is 1, then A has an abundance of soft elements, and (Cu(A), ΣA) is ideal-filtered precisely when A has an abundance of soft elements and Cu(A) is ideal-filtered. A similar result holds for unital A with covering number 1. We prove that if Λ is a collection of unital nuclear C*-algebras for which the tracial covering number does not exceed m, then any simple unital C*-algebra A that falls within the corresponding collection of weakly tracially approximated C*-algebras will also possess a tracial covering number of at most m. Similar conclusions are derived for the tracial approximate covering number and the tracial unit approximate covering number, and finally, we explore the relationships among these three types of tracial covering numbers.