Stable rank of corner rings

Author:

Ara P.,Goodearl K.

Abstract

B. Blackadar recently proved that any full corner p A p pAp in a unital C*-algebra A A has K-theoretic stable rank greater than or equal to the stable rank of A A . (Here p p is a projection in A A , and fullness means that A p A = A ApA=A .) This result is extended to arbitrary (unital) rings A A in the present paper: If p p is a full idempotent in A A , then sr ( p A p ) sr ( A ) \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 p A q pAq . The main result yields estimates relating stable ranks of Morita equivalent rings. In particular, if B End A ( P ) B\cong \operatorname {End}_{A}(P) where P A P_{A} is a finitely generated projective generator, and P P can be generated by n n elements, then sr ( A ) n sr ( B ) n + 1 \operatorname {sr}(A)\le n{\cdot }\operatorname {sr}(B)-n+1 .

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference10 articles.

1. The almost isomorphism relation for simple regular rings;Ara, Pere;Publ. Mat.,1992

2. A stable cancellation theorem for simple 𝐶*-algebras;Blackadar, Bruce;Proc. London Math. Soc. (3),1983

3. The stable rank of full corners in 𝐶*-algebras;Blackadar, Bruce;Proc. Amer. Math. Soc.,2004

4. The stable range of 𝐶*-algebras;Herman, Richard H.;Invent. Math.,1984

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