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Unimodular matrices in Banach algebra theory


Authors: Gustavo Corach and Angel R. Larotonda
Journal: Proc. Amer. Math. Soc. 96 (1986), 473-477
MSC: Primary 46H05; Secondary 46M20
DOI: https://doi.org/10.1090/S0002-9939-1986-0822443-7
MathSciNet review: 822443
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Abstract: Let $ A$ be a ring with 1 and denote by $ L$ (resp. $ R$) the set of left (resp. right) invertible elements of $ A$. If $ A$ has an involution *, there is a natural bijection between $ L$ and $ R$. In general, it seems that there is no such bijection; if $ A$ is a Banach algebra, $ L$ and $ R$ are open subsets of $ A$, and they have the same cardinality. More generally, we prove that the spaces $ {U_k}({A^n})$ of $ n \times k$-left-invertible matrices and $ _kU({A^n})$ of $ k \times n$-right-invertible matrices are homotopically equivalent. As a corollary, we answer negatively two questions of Rieffel [12].


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DOI: https://doi.org/10.1090/S0002-9939-1986-0822443-7
Article copyright: © Copyright 1986 American Mathematical Society

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