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Zbl 1242.46066
Wootijirattikal, Titarii; Ong, Sing-Cheong
Functional decomposition of state induced $C^\ast$-matrix spaces. (English)
[J] Banach J. Math. Anal. 5, No. 2, 106-121, electronic only (2011). ISSN 1735-8787/e

Summary: 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 $f = g + h$ . 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 ${\mathit tr}(AB) = {\mathit tr}(BA)$, without using the operator theoretical machineries used in the known proofs.

MSC 2000:: *46L05 General theory of C*-algebras
46L30 States of C*-algebras

Keywords: $C^\ast$-algebra; state space; weak topology; dual space

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