Unimodular matrices in Banach algebra theory

Corach, Gustavo; Larotonda, Angel R.

doi:10.1090/S0002-9939-1986-0822443-7

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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