We characterize the $C^\star$-algebras for which openness of projections in their second duals is preserved under Murray-von Neumann equivalence. They are precisely the extensions of the annihilator $C^\star$-algebras by the commutative $C^\star$-algebras. We also show that the annihilator $C^\star$-algebras are precisely the $C^\star$-algebras for which all projections in their second duals are open.
- Pub Date:
- April 2019
- Mathematics - Operator Algebras;
- Mathematics - Functional Analysis;
- 47L50 (Primary) 46L10;
- 46M05 (Secondary)
- 11 pages