# Open projections and Murray-von Neumann equivalence

## TL;DR

This paper characterizes specific $C^*$-algebras based on the openness of projections in their second duals and their relation to Murray-von Neumann equivalence, identifying extensions of annihilator algebras by commutative algebras.

## Contribution

It provides a complete characterization of $C^*$-algebras where projection openness is preserved under Murray-von Neumann equivalence, linking it to extensions of annihilator algebras.

## Key findings

- Identifies $C^*$-algebras as extensions of annihilator algebras by commutative algebras.
- Shows annihilator $C^*$-algebras have all projections in their second duals open.
- Establishes a precise condition for the preservation of projection openness under equivalence.

## Abstract

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.