Stable rank of corner rings

Abstract:

B. Blackadar recently proved that any full corner $pAp$ in a unital C*-algebra $A$ has K-theoretic stable rank greater than or equal to the stable rank of $A$. (Here $p$ is a projection in $A$, and fullness means that $ApA=A$.) This result is extended to arbitrary (unital) rings $A$ in the present paper: If $p$ is a full idempotent in $A$, then $\operatorname {sr} (pAp)\ge \operatorname {sr}(A)$. The proofs rely partly on algebraic analogs of Blackadar’s methods and partly on a new technique for reducing problems of higher stable rank to a concept of stable rank one for skew (rectangular) corners $pAq$. The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if $B\cong \operatorname {End}_{A}(P)$ where $P_{A}$ is a finitely generated projective generator, and $P$ can be generated by $n$ elements, then $\operatorname {sr}(A)\le n{\cdot }\operatorname {sr}(B)-n+1$.