An operator algebra weight on a C*-algebra 
is a map ![phi:A_+->[0,infty]](/images/equations/OperatorAlgebraWeight/Inline2.svg)
on its positive cone such that 
, 
, and 
for 
and 
. A weight is lower semicontinuous in the norm topology
when its sublevel sets are closed. On a von Neumann
algebra, it is normal when it preserves suprema of bounded increasing nets of
positive elements (Haagerup 1975).
Blecher and Choi (2026) proved that a weight is lower semicontinuous iff its restrictions to all commutative subalgebras of 
are lower semicontinuous. For a von
Neumann algebra, they proved that a weight is normal iff
its restrictions to all commutative von Neumann subalgebras are normal. No faithfulness
or semifiniteness hypothesis is required. Blecher and Choi (2026) report AI assistance
with the mathematical proof.
-Algebras." J. Funct. Anal. 19, 302-317,
1975.