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Jul 30, 2013 at 15:21 comment added Gabor Szabo Another simple-minded example in this vein would be almost anything involving $A=\mathcal{O}_2$. There are many automorphism whose fixed point algebra has non-trivial K-theory. While every pair of non-trivial projections in $\mathcal{O}_2$ is MvN equivalent, this is certainly not true for every pair of non-trivial projections in most fixed point algebras you encounter.
Jul 30, 2013 at 6:07 history answered Michael CC BY-SA 3.0