$\mathrm{C}^{*}$-algebras hereditarily containing nonzero, square-zero elements

Abstract We introduce and study the weak Glimm property for $\mathrm{C}^{*}$ -algebras, and also a property we shall call (HS $_0$ ). We show that the properties of being nowhere scattered and residual (HS $_0$ ) are equivalent for any $\mathrm{C}^{*}$ -algebra. Also, for a $\mathrm{C}^{*}$ -algebra with the weak Glimm property, the properties of being purely infinite and weakly purely infinite are equivalent. It follows that for a $\mathrm{C}^{*}$ -algebra with the weak Glimm property such that the absolute value of every nonzero, square-zero, element is properly infinite, the properties of being (weakly, locally) purely infinite, nowhere scattered, residual (HS $_0$ ), residual (HS $_{\text {t}}$ ), and residual (HI) are all equivalent, and are equivalent to the global Glimm property. This gives a partial affirmative answer to the global Glimm problem, as well as certain open questions raised by Kirchberg and Rørdam.