On a counterexample to a conjecture by Blackadar

TL;DR

This paper discusses a counterexample to Blackadar's conjecture on semiprojectivity in split short-exact sequences, showing how to modify existing examples to find a non-semiprojective algebra with a semiprojective ideal.

Contribution

The authors demonstrate how to modify previous counterexamples to produce a non-semiprojective algebra with a semiprojective ideal where the quotient map does not split.

Findings

Counterexamples exist where the algebra is not semiprojective despite having a semiprojective ideal.

The quotient map from the algebra to the complex numbers can be non-split in these counterexamples.

Modification of existing examples can produce new counterexamples with desired properties.

Abstract

Blackadar conjectured that if we have a split short-exact sequence 0 -> I -> A -> A/I -> 0 where I is semiprojective and A/I is isomorphic to the complex numbers, 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 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.