Is there a non-commutative non-unital Banach algebra $A$ for which $aa_0 -a_{0}a$ lies in the annihilator of $A$ for any $a\in A$? Here $a_0$ is an element of $A$ not belonging to its centre $Z(A)$.
Could you please suggest me a good reference (on Banach algebras) including examples like this?
Any help is appreciated.