Unimodular matrices in Banach algebra theory

Abstract

Let A A be a ring with 1 and denote by L L (resp. R R ) the set of left (resp. right) invertible elements of A A . If A A has an involution *, there is a natural bijection between L L and R R . In general, it seems that there is no such bijection; if A A is a Banach algebra, L L and R R are open subsets of A A , and they have the same cardinality. More generally, we prove that the spaces U k ( A n ) {U_k}({A^n}) of n × k n \times k -left-invertible matrices and k U ( A n ) _kU({A^n}) of k × n k \times n -right-invertible matrices are homotopically equivalent. As a corollary, we answer negatively two questions of Rieffel [12].