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Strict Comparison of Positive Elements in Multiplier Algebras
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- Journal:
- Canadian Journal of Mathematics / Volume 69 / Issue 2 / 01 April 2017
- Published online by Cambridge University Press:
- 20 November 2018, pp. 373-407
- Print publication:
- 01 April 2017
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- Article
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Main result: If a
${{C}^{*}}$-algebra 
$\mathcal{A}$ is simple, 
$\sigma $-unital, has finitely many extremal traces, and has strict comparison of positive elements by traces, then its multiplier algebra 
$\mathcal{M}\left( \mathcal{A} \right)$ 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. As an application of strict comparison, if $\mathcal{A}$ 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.
