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

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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 ??

## marked as duplicate by José Carlos Santos, Saad, Arnaud Mortier, Ethan Bolker, Xander HendersonMar 31 '18 at 0:02

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