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

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 $A$ be an arbitrary $C^*$ algebra. In \cite{BH} Blackadar and Handelman conjectured the set of lower semicontinuous dimension functions on $A$ to be pointwise dense in the set $DF(A)$ of all dimension functions on $A$ and $DF(A)$ to be a Choquet simplex. We provide an equivalent condition for the first conjecture for unital $A$. Then by applying this condition we confirm the first Conjecture for all unital $A$ for which either the radius of comparison is finite or the semigroup $W(A)$ 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 $A$ and extreme boundary of the the quasitrace simplex of $A$ is finite, then every dimension function of $A$ is lower semicontinuous and $DF(A)$ is affinely homeomorphic to the quasitrace simplex of $A$. Combing this with the said results on the first Conjecture give us a class of algebras for which $DF(A)$ is a Choquet simplex, i.e. gives a new class for which the 2nd Conjecture mentioned above holds.