Positive combinations of projections in von Neumann algebras and purely infinite simple \(C^*\)-algebras (Q424889)

This paper is a survey on the following interesting questions involving projections in \(C^*\)-algebras: (i) Which positive elements in a \(C^*\)-algebra are linear combination of projections with positive coefficients? And for which algebras are all positive elements positive combinations of projections? (ii) Which positive elements in a \(C^*\)-algebra are finite sums of projections? (iii) Which positive operators in a \(C^*\)-algebra are finite or infinite sums of projections? The authors study these questions for von Neumann algebras, for \(\sigma\)-unital purely infinite simple \(C^*\)-algebras and for their associated multiplier algebras.