Taiwanese Journal of Mathematics

RANK PRESERVING IN INTEGRAL EXTENSIONS OF COMMUTATIVE $C^*$-ALGEBRAS

Chung-Wen Tsai and Ngai-Ching Wong

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Abstract

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 $\operatorname{tsr}(B) = 1$ whenever $\operatorname{tsr}(A) = 1$. In this note, we investigate whether we will have $\operatorname{tsr}(A) = \operatorname{tsr}(B)$ in general. For instance, when $A$ is a commutative unital $C^*$-algebra, we show that $\operatorname{tsr}(A) \leq \operatorname{tsr}(B)$, and the equality holds at least when the integral extension is separable. In general, $A$ and $B$ have the same Bass stable ranks $\operatorname{Bsr}(A) = \operatorname{Bsr}(B)$.

Article information

Source
Taiwanese J. Math., Volume 16, Number 2 (2012), 545-553.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406601

Digital Object Identifier
doi:10.11650/twjm/1500406601

Mathematical Reviews number (MathSciNet)
MR2892898

Zentralblatt MATH identifier
1252.46041

Subjects
Primary: 46J10: Banach algebras of continuous functions, function algebras [See also 46E25] 46L05: General theory of $C^*$-algebras

Keywords
topological stable rank Bass stable ranks covering dimension integral extension

Citation

Tsai, Chung-Wen; Wong, Ngai-Ching. RANK PRESERVING IN INTEGRAL EXTENSIONS OF COMMUTATIVE $C^*$-ALGEBRAS. Taiwanese J. Math. 16 (2012), no. 2, 545--553. doi:10.11650/twjm/1500406601. https://projecteuclid.org/euclid.twjm/1500406601


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