Strong sums of projections in von Neumann factors

Abstract

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.

Commutators and linear spans of projections in certain finite C*-algebras

“…An analogous result holds in σ−finite type II ∞ von Neumann factors where we proved in [26,Corollary 3.5] that all positive elements are positive combinations of projections except those that have infinite range projection and belong to the Breuer ideal generated by all finite projections. Moreover, all positive elements in a von Neumann factor of type I n , II 1 , or σ−finite type III are positive combinations of projections ( [26,Theorem 2.12]).…”

“…An analogous result holds in σ−finite type II ∞ von Neumann factors where we proved in [26,Corollary 3.5] that all positive elements are positive combinations of projections except those that have infinite range projection and belong to the Breuer ideal generated by all finite projections. Moreover, all positive elements in a von Neumann factor of type I n , II 1 , or σ−finite type III are positive combinations of projections ( [26,Theorem 2.12]). In the non σ−finite case or in von Neumann algebras with a nontrivial center, a necessary and sufficient condition for a positive element to be a positive combination of projections is given in terms of central ideals and the central essential spectrum ( [26,Theorem 2.12]).…”

“…We will present the proof through the chain of the following lemmas. Our first result extends to the present setting the main tool that we used in [25] and [26]. Proof.…”

“…Moreover, all positive elements in a von Neumann factor of type I n , II 1 , or σ−finite type III are positive combinations of projections ( [26,Theorem 2.12]). In the non σ−finite case or in von Neumann algebras with a nontrivial center, a necessary and sufficient condition for a positive element to be a positive combination of projections is given in terms of central ideals and the central essential spectrum ( [26,Theorem 2.12]).…”

Positive combinations and sums of projections in purely infinite simple 𝐶*-algebras and their multiplier algebras

“…The weaker problem when the sums are allowed to be infinite has been completely solved when C = B(H), C is a type III factor and for diagonalizable operators when C is a type II factor (with convergence in the strong operator topology) [10], and also when C is the multiplier algebra of a simple purely infinite σ-unital but nonunital C * -algebra (with convergence in the strict topology) [11]. Our main results are that every positive element of a simple purely infinite σ-unital C * -algebra A is a positive combination of projections (Theorem 2.11).…”