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arXiv · 1210.4756

On a counterexample to a conjecture by Blackadar

Adam P. W. Sørensen
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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.

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Adam P. W. Sørensen. 2012-10-17. On a counterexample to a conjecture by Blackadar. https://arxiv.org/abs/1210.4756

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