Ranks of Operators in Simple C ∗ -algebras with Stable Rank One
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TLDR
In particular, the Toms-Winter conjecture holds for simple, approximately subhomogeneous, stable algebras with stable rank one as mentioned in this paper, assuming that A is stable if and only if it has strict comparison of positive elements.Abstract:
Let A be a simple $$C^*$$-algebra with stable rank one. We show that every strictly positive, lower semicontinuous, affine function on the simplex of normalized quasitraces of A is realized as the rank of an operator in the stabilization of A. Assuming moreover that A has locally finite nuclear dimension, we deduce that A is $$\mathcal {Z}$$-stable if and only if it has strict comparison of positive elements. In particular, the Toms–Winter conjecture holds for simple, approximately subhomogeneous $$C^*$$-algebras with stable rank one.read more
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