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$\newcommand{\odd}{\mathrm{odd}}\newcommand{\even}{\mathrm{even}}$Let $X$ be a complex manifold, you can assume it's compact, if necessary. We have the Dolbeault complex $$0 \rightarrow \mathcal{A}^{0,...

Let $A$ be the UHF-algebra of type $2^{\infty}$. Suppose that $p$ and $q$ are two projections in $A$ and $\tau(p) = \tau(q)$, where $\tau$ is the unique normalized trace. Then there is a partail ...