This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum converging in the strong operator topology if the collection is infinite. A similar necessary condition is given when the operator and the projections are taken in a type II von Neumann factor, and the condition is proven to be also sufficient if the operator is "diagonalizable". A simpler necessary and sufficient condition is given in the type III factor case.

Abstract.Main result: If a C*-algebra A is simple, σ-unital, has nitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra M(A) also has strict comparison of positive elements by traces.e same results holds if " nitely many extremal traces" is replaced by "quasicontinuous scale". A key ingredient in the proof is that every positive element in the multiplier algebra of an arbitrary σ-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If A is a simple separable stable C*-algebra with real rank zero, stable rank one, and strict comparison of positive elements by traces, then whether a positive element is a positive linear combination of projections is determined by the trace values of its range projection.

Abstract. Every positive element in a purely infinite simple σ-unital C * -algebra A is a finite linear combination of projections with positive coefficients. Also, every positive a in the multiplier algebra M(A) of a purely infinite simple σ-unital C * -algebra A is a finite linear combination of projections with positive coefficients. Furthermore, if the essential norm a ess > 1, then a is a finite sum of projections in M(A). As a consequence, any positive element in the generalized Calkin Algebra M(A)/A or in M(A) but not in A is a positive scalar multiple of a finite sum of projections.

In this paper, we study the problem of when the corona algebra of a nonunital C * -algebra is purely infinite. A complete answer is obtained for stabilisations of simple and unital algebras that have enough comparison of positive elements. Our result relates the pure infiniteness condition (from its strongest to weakest forms) to the geometry of the tracial simplex of the algebra, and to the behaviour of corona projections, despite the fact that there is no real rank zero condition. IntroductionThe corona is a noncommutative generalization of the famous Stone-Čech corona of topological spaces. It is well-known that properties of the corona have deep

Abstract. We first prove that in a σ-finite von Neumann factor M , a positive element a with properly infinite range projection Ra is a linear combination of projections with positive coefficients if and only if the essential norm a e with respect to the closed two-sided ideal J(M ) generated by the finite projections of M does not vanish. Then we show that if a e > 1, then a is a finite sum of projections. Both these results are extended to general properly infinite von Neumann algebras in terms of central essential spectra.Secondly, we provide a necessary condition for a positive operator a to be a finite sum of projections in terms of the principal ideals generated by the excess part a + := (a − I)χa(1, ∞) and the defect part a − := (I − a)χa(0, 1) of a; this result appears to be new also for B(H).Thirdly, we prove that in a type II 1 factor a sufficient condition for a positive diagonalizable operators to be a finite sum of projections is that τ (a + ) − τ (a − ) > 0.