Projection decomposition in multiplier algebras

Abstract

In this paper we present new structural information about the multiplier algebra Mult (A) of a sigma-unital purely infinite simple C*-algebra A, by characterizing the positive elements a in Mult(A) that are strict sums of projections belonging to A. If a is not in A and is not a projection, then the necessary and sufficient condition for a to be a strict sum of projections belonging to A is that the norm ||a||>1 and that the essential norm ||a||_ess >=1. Based on a generalization of the Perera-Rordam weak divisibility of separable simple C*-algebras of real rank zero to all sigma-unital simple C*-algebras of real rank zero, we show that every positive element of A with norm greater than 1 can be approximated by finite sums of projections. Based on block tri-diagonal approximations, we decompose any positive element a in Mult(A) with ||a||>1 and ||a||_ess >=1 into a strictly converging sum of positive elements in A with norm greater than 1.

To appear in Mathematische Annalen

References in corpus (2)

• Frames in Hilbert C*-modules and C*-algebras
• A module frame concept for Hilbert C*-modules

Cited by in corpus (7)

• Unbounded quasitraces, stable finiteness and pure infiniteness
• Finite sums of projections in von Neumann algebras
• Sums of equivalent sequences of positive operators in von Neumann factors
• Commutators and linear spans of projections in certain finite C*-algebras
• Strong sums of projections in type $\mathrm{II}$ factors
• On finite sums of projections and Dixmier's averaging theorem for type ${\mathrm{II}}_1$ factors
• Finite sums of projections in purely infinite simple C*-algebras with torsion K_0