1 results
$\mathrm{C}^{*}$-algebras hereditarily containing nonzero, square-zero elements
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 04 November 2024, pp. 1-11
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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.
