On a Counterexample to a Conjecture by Blackadar

Blackadar conjectured that if we have a split short-exact sequence 0→I→A→C→0 where I is semiprojective then A must be semiprojective. Eilers and Katsura have found a counterexample to this conjecture. Presumably Blackadar asked that the extension be split to make it more likely that semiprojectivity of I would imply semiprojectivity of A. But oddly enough, in all the counterexamples of Eilers and Katsura the quotient map from A to A/I≅C is split. We will show how to modify their examples to find a non-semiprojective C∗-algebra B with a semiprojective ideal J such that B∕J is the complex numbers and the quotient map does not split.