a separable $c^*$-algebra $a$ is said to be $\ell$-open ( or $\ell$-closed) when the image of hom(a, b) is open (or closed) in hom(a, b/i), for all separable $c^*$-algebras b and ideals i. the concept of semiprojectivity has been used many times in the classification of c*-algebras. bruce blackadar introduced $\ell$-open and $\ell$-closed $c^*$-algebras as a superclass of semiprojective $c^*$-algebras.

in recent work with a. tikuisis, we characterize $\ell$-open and $\ell$-closed $c^*$-algebras and deduce that $\ell$-open $c^*$-algebras are $\ell$-closed as conjectured by blackadar. moreover, we show that the notion of $\ell$-open $c^*$-algebras and semiprojective $c^*$-algebras coincide for commutative unital $c^*$-algebras.