Let A be simple separable unital C*-algebra, and let B be a large subalgebra of A. If B has stable rank one, then so does A.
We conjecture that if X is a compact metrizable space, and h is a minimal homeomorphism of X, then the associated transformation group C*-algebra has stable rank one, regardless of the mean dimension of h (that is, even if this algebra is not expected to be classifiable in the sense of the Elliott program). Using the stable rank result above, we can prove this conjecture in some cases.
This work is joint with Dawn Archey, and the conjecture is also made jointly with Zhuang Niu.