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Journal of Operator Theory

Volume 90, Issue 2, Autumn 2023  pp. 331-364.

General and connected stable ranks of some Banach algebras of continuous functions

Authors:  Kazuhiro Kawamura
Author institution: Department of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan

Summary:  We take a topological approach to the study of general and connected stable ranks, introduced by Rieffel, of the Banach algebras of Banach algebra-valued continuous functions on compact Hausdorff spaces. We first study a result of Vaidyanathan and its variant on the general stable rank. Next we pursue the research initiated by Nica and Vaidyanathan to find conditions under which the general/connected stable rank of a commutative $C^\ast$-algebra over a compact CW-complex attains its maximum value. Some further examples are discussed as an illustration of our method.

Keywords: stable rank, homotopical stable rank, homotopy group of spheres and unitary groups, fibration

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