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

Amini, Massoud  ; Elliott, George A. ; Rouzbehani, Mohammad 

Canadian mathematical bulletin, Tome 67 (2024) no. 4, pp. 1081-1091

Résumé

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.

Détail

DOI : 10.4153/S0008439524000511

Mots-clés : C*-algebra, square-zero elements, finiteness properties