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I asked this question (https://math.stackexchange.com/questions/3076735/an-example-of-a-banach-algebra-satisfying-given-conditions) but unfortunately no one answered it. Please help me to find an example of a Banach algebra ( if any) with the following property:

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.

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    $\begingroup$ Take the algebra of upper-triangular $3\times 3$ real matrices with $0$ on the diagonal and any non-central $a_0$. More generally, any non-commutative Banach algebra $A$ with $A^3=0$ will work. $\endgroup$
    – Uriya First
    Commented Jan 17, 2019 at 21:12
  • $\begingroup$ Thank you very much @Uriya First this simple (and nice of course) example works. $\endgroup$
    – Fermat
    Commented Jan 17, 2019 at 21:45
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    $\begingroup$ Just to reply to the particular wording of the question: it is far more valuable to learn how to think about (Banach) algebras using basic logic and invention than to seek a ready-made list of counterexamples in "a good reference". Basically, if you cannot prove "all Banach algebras have property X", examine where your proof breaks down, and you may be led to construct a counterexample by yourself $\endgroup$
    – Yemon Choi
    Commented Jan 18, 2019 at 12:28

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