New results on common properties of the products AC and BA, II.

In this note, we continue to investigate common properties of the products in the setting of rings, bounded linear operators, or Banach algebras. We prove: (i) If a,b,c are elements in a unital associative ring R satisfying aba=aca, then von Neumann regularity (resp. generalized Fredholmness relative to an ideal I of R) of 1−ac is converted into that of 1−ba. (ii) If A,B,C are bounded linear operators satisfying ABA=ACA, then I−AC and I−BA share common complementability of kernels and ranges. (iii) If a,b,c are elements in a unital semisimple Banach algebra A satisfying aba=aca and I is a trace ideal of A such that soc(A)⊆I⊆ kh(soc(A)), then 1−ac and 1−ba share common Fredholmness relative to I and have the same abstract index. A similar result holds for B‐Fredholmness in primitive Banach algebra. [ABSTRACT FROM AUTHOR]