Strict comparison of positive elements in multiplier algebras

TL;DR

This paper proves that strict comparison of positive elements by traces extends from a simple, $\sigma$-unital C*-algebra to its multiplier algebra under certain conditions, with applications to projections and trace values.

Contribution

It establishes the extension of strict comparison of positive elements to multiplier algebras for a broad class of C*-algebras, introducing bi-diagonal series approximation techniques.

Findings

Strict comparison extends to multiplier algebras under specified conditions.

Positive elements can be approximated by bi-diagonal series.

Trace-based criteria determine when positive elements are linear combinations of projections.

Abstract

Main result: If a C*-algebra is simple, $\sigma$-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier also has strict comparison of positive elements by traces. The same results holds if "finitely 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 $\sigma$-unital C*-algebra can be approximated by a bi-diagonal series. An application of strict comparison: If the algebra is a simple separable stable $\sigma$-unital with real rank zero, stable rank one, strict comparison of positive elements by traces, then whether a positive element is a linear combination of projections depends on the trace values of its range projection.