Rank Preserving in Integral Extensions of Commutative C∗-algebras
نویسندگان
چکیده
Let A, B be two regular commutative unital Banach algebras such that B is integral over A. In 2003, Dawson and Feinstein showed that the topological stable rank tsr(B) = 1 whenever tsr(A) = 1. In this note, we investigate whether we will have tsr(A) = tsr(B) in general. For instance, when A is a commutative unital C∗-algebra, we show that tsr(A) ≤ tsr(B), and the equality holds at least when the integral extension is separable. In general, A and B have the same Bass stable ranks Bsr(A) = Bsr(B).
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