if $a,b$ are elements of a unital algebra $A,$ then $1-ab$ is invertible if and only if $1-ba$ is invertible.

Question

if $a,b$ are elements of a unital algebra $A,$ then $1-ab$ is invertible if and only if $1-ba$ is invertible.

because if $1-ab\ $ has inverse $x$ , then $1-ba\ $ has inverse $1+bxa$. but how ??

$$(1-ab)x=x(1-ab)=1$$

then $$(1-ba)(1+bxa)=1+bxa-ba-babxa$$

but how the expression on right hand side equal to $1$.

any hint ??

Answer 1

From $(1-ab)x=1$ we get $abx=x-1$. This and $(1-ba)(1+bxa)=1+bxa-ba-b(abx)a$ give

$(1-ba)(1+bxa)=1$.