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A nonsmooth exponential
Let ${\cal M}$ be a type II$_1$ von Neumann algebra, $\tau $ a trace in ${\cal M}$, and $L^2({\cal M},\tau )$ the GNS Hilbert space of $\tau $. If $L^2({\cal M},\tau )_+$ is the completion of the set ${\cal M}_{\rm sa}$ of selfadjoint elements, then each