Show that a subalgebra is commutative.

Question

If $B$ is an unital algebra (even not commutative), how do I show that the subalgebra spanned by the elements $1$, $f$ and $(f - \lambda1)^{-1}$ is commutative? Thank you.

Answer 1

It suffices to show that the generators commute and for that you just need to show that $f$ and $(f - \lambda1)^{-1}$ commute with each other ($1$ commutes with everything). Well, note that $f$ and $f - \lambda1$ commute and if $x$, and $y$ commute then $x$ and $y^{-1}$ commute: $$xy^{-1} = y^{-1}yxy^{-1} = y^{-1}xyy^{-1} = y^{-1}x$$