Strict comparison for a 
-algebra is an order-detection property of its Cuntz
semigroup using rank functionals. In the extended-functional convention, if 
, the hypothesis is 
for every semigroup functional 
, with strict inequality
whenever 
.
Strict comparison requires that these conditions imply 
. This convention retains zero and infinite
target values. For a simple exact unital algebra the normalized-trace formulation
compares positive 
with 
when 
for every tracial state.
A C*-algebra is nonelementary when it is not isomorphic to the compact operators on a Hilbert space.
-Algebras:
99 Problems." 12 Jun 2025.