Projection decomposition in multiplier algebras

TL;DR

This paper characterizes positive elements in the multiplier algebra of certain C*-algebras as sums of projections, providing new structural insights and approximation methods for these elements.

Contribution

It introduces a characterization of positive elements as strict sums of projections and generalizes weak divisibility to all sigma-unital simple C*-algebras of real rank zero.

Findings

Positive elements with norm > 1 can be approximated by finite sums of projections.

Decomposition of elements in Mult(A) into sums of positive elements in A.

Established conditions for elements to be strict sums of projections.

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.