On $\mathrm{\ell }$-open $C^{\ast }$ algebras and $\mathrm{\ell }$-closed $C^{\ast }$-algebras

In this paper, we characterize $\mathrm{\ell }$-open and $\mathrm{\ell }$-closed $C^{\ast }$-algebras and deduce that $\mathrm{\ell }$-open $C^{\ast }$-algebras are $\mathrm{\ell }$-closed, as conjectured by Blackadar. Moreover, we show that a commutative unital $C^{\ast }$-algebra is $\mathrm{\ell }$-open if and only if it is semiprojective.