In the literature, are there some research around a directed graph associated to a $C^*$ algebra or a ring $A$ whose vertices are projections or idempotents of $A$ and $e$ is connected to $f$ iff $ef=fe=f$? What properties of this graph reflect some part of structure of $A$?
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asked Dec 30, 2019 at 14:49
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Please check out this book:
Goodearl, Kenneth R. Von Neumann regular rings. Vol. 4. London: Pitman, 1979.
I'm pretty sure I remember this being discussed (but in terms of a partial ordering, which you can re-interpret as a digraph).
I do not recall specifically projections being analyzed in terms of this partial order, because the book isn't specifically about $C^\ast$ algebras, but it should be general enough to cover the basics, and informative enough to point you toward the specifics.
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