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AbelianPairwiseorthogonal projections in C*-algebras

Recall that a

Is every non-zero projection $p$ in a C*-algebra $A$ is abelian if the C*-algebra $pAp$ is abelian. Does every C*-algebra admit an abelian non-zero projection?

Is every non-projection a limit supremum or infimum (at least majorized by / majorizes) a family of pairwise orthogonal non-zero projections ?in $A$?

PS. Are there any cheap ways to generate abelian sub C*-algebras of a given C*-algebra (of course, having in mind Akemann's example of a non-separable C*-algebra with each abelian sub-C*-algebra separable)?

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Recall that a projection $p$ in a C*-algebra $A$ is abelian if the C*-algebra $pAp$ is abelian. Does every C*-algebra admit an abelian non-zero projection?

Is every non-projection a limit of pairwise orthogonal projections?

PS. Are there any cheap ways to generate abelian sub C*-algebras of a given C*-algebra (of course, having in mind Akemann's example of a non-separable C*-algebra with each abelian sub-C*-algebra separable)?

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Abelian projections in C*-algebras

Recall that a projection $p$ in a C*-algebra $A$ is abelian if the C*-algebra $pAp$ is abelian. Does every C*-algebra admit an abelian projection?

PS. Are there any cheap ways to generate abelian sub C*-algebras of a given C*-algebra (of course, having in mind Akemann's example of a non-separable C*-algebra with each abelian sub-C*-algebra separable)?