I asked this question

(Https://math.stackexchange.com/questions/3076735/an-example-of-a-banach-algebra-satisfying-given-conditions)

but unfortunately nobody answered it. Please, help me find an example of Banach algebra (if any) with the following property:

Non commutative noncommutative Banach algebra $ A $ For who $ aa_0 -a_ {0} a $ is in the annihilator of $ A $ for all $ a in A $.

Right here $ a_0 $ is an element of $ A $ not belonging to his center $ Z (A) $.

Could you please suggest me a good reference (on Banach algebras) with examples like this?

Any help is appreciated.