## Unimodular matrices in Banach algebra theory

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- by Gustavo Corach and Angel R. Larotonda
- Proc. Amer. Math. Soc.
**96**(1986), 473-477 - DOI: https://doi.org/10.1090/S0002-9939-1986-0822443-7
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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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## Bibliographic Information

- © Copyright 1986 American Mathematical Society
- 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