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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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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Article copyright: © Copyright 1986 American Mathematical Society