Abstract:
On a von Neumann algebra $M$, we consider traces with values in the algebra $L^0$ of measurable complex-valued functions. We show that every faithful normal $L^0$-valued trace on $M$ generates an $L^0$-valued metric on the algebra of measurable operators that are affiliated with $M$. Moreover, convergence in this metric coincides with local convergence in measure.
Key words:von Neumann algebra, measurable operator, local convergence in measure, vectorvalued trace.
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