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A solution to Haagerup's problem: Normality of weights is commutatively determined
We prove that a weight on a C$^*$-algebra is lower semi-continuous if it is lower semi-continuous on commutative C$^*$-subalgebras. This can be used to prove a von Neumann algebraic counterpart that a weight on a von Neumann algebra is normal if it is normal on commutative von Neumann subalgebras. T
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