Stable range in C*-algebras

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Abstract
A unital C*-algebra A is said to have unitary 1-stable range (8) if for all pairs a, b of elements of A satisfying aA + bA = A there exists a unitary u in A such that a + bu is invertible. This concept is somewhat stronger than the usual stable range condition of algebraic K-theory ((3), chapter V). Handelman(8) shows among other things that finite AW*-algebras have unitary 1-stable range and uses this fact to study the algebraic K1 of a finite AW*-algebra. We prove below that a unital C*-algebra has unitary 1-stable range if and only if its group of invertible elements is dense. In addition we give some consequences of this fact and consider the related question of (unitary) polar decomposition in C*-algebras.

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