Abstract
A theorem of Dixmier states that each bounded linear functional $f$ on the algebra of bounded linear operators on a separable Hilbert space is a direct sum of a trace functional $g$ and a singular functional $h$, vanishing on the compact operators, such that $\Vert f \Vert= \Vert g \Vert +\Vert h \Vert$. We use elementary methods to construct, via the state space of a $C^\ast$-algebra, a Banach space of $C^\ast$ matrices that contains a closed subspace on which a version of Dixmier's theorem is proved. When the $C^\ast$-algebra is taken to be the complex numbers our approach gives elementary and transparent proofs of Dixmier's theorem and the trace formula $\rm{tr}(AB) = \rm{tr}(BA)$, without using the operator theoretical machineries used in the known proofs.
Citation
Sing-Cheong Ong. Titarii Wootijirattikal. "Functional decomposition of state induced C*-matrix spaces." Banach J. Math. Anal. 5 (2) 106 - 121, 2011. https://doi.org/10.15352/bjma/1313363006
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