The extended trace cone 
of a 
-algebra consists of all lower semicontinuous maps ![tau:A_+->[0,infty]](/images/equations/ExtendedTraceCone/Inline3.svg)
that vanish at zero, are additive and
positively homogeneous, and satisfy 
. The finite
domain is not required to be dense. Its canonical topology is characterized by
 |
where 
and 
. The positive cut 
uses continuous functional
calculus. Every closed two-sided ideal 
contributes a trace taking value zero
on 
and infinity on positive elements outside 
. Thus the cone retains ideal information in addition
to finite trace values.
-Algebras:
99 Problems." 12 Jun 2025.