Functional decomposition of state induced C*-matrix spaces

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.