Elements with a soft Cuntz class appear naturally when trying to realize every functional on the Cuntz semigroup as a rank map, a deep and open problem closely related to the Toms-Winter conjecture.

Using the definition of Cu-softness as an inspiration, I will define a C*-algebraic version of softness and examine the relation between both notions.

I will then focus on C*-algebras that contain an abundance of positive soft elements. These algebras can be characterized in terms of properties of their Cuntz semigroup, and this characterization is in turn related to the Global Glimm Problem, which will be the focus of the third session of the talk.