A note on two Conjectures on Dimension funcitons of $C^{*}$-algebras
Kaushika De Silva

TL;DR
This paper investigates conjectures about the structure of dimension functions on $C^*$-algebras, providing new conditions under which these conjectures hold, especially for unital algebras with certain properties.
Contribution
It offers an equivalent condition for the first conjecture, confirms it for specific classes of unital $C^*$-algebras, and links the structure of dimension functions to the quasitrace simplex.
Findings
Confirmed the first conjecture for unital $A$ with finite radius of comparison or almost unperforated $W(A)$.
Established that $DF(A)$ is a Choquet simplex under certain conditions.
Connected the structure of dimension functions to the quasitrace simplex for specific classes of algebras.
Abstract
\noindent Let be an arbitrary algebra. In \cite{BH} Blackadar and Handelman conjectured the set of lower semicontinuous dimension functions on to be pointwise dense in the set of all dimension functions on and to be a Choquet simplex. We provide an equivalent condition for the first conjecture for unital . Then by applying this condition we confirm the first Conjecture for all unital for which either the radius of comparison is finite or the semigroup is almost unperforated. As far as we know the most general results on the first Conjecture up to now assumes exactness, simplicity and moreover stronger regularity properties such as strict comparison. Our results are achieved through applications of the techniques developed in \cite{BR} and \cite{R}. We also note that, whenever the first Conjecture holds for some unital and extreme…
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Advanced Banach Space Theory
