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Let $A$ be a Banach algebra. The center of $A$, denoted by $Z(A)$, is the set of elements of $A$ that commute with all elements of $A$. Please give some examples of Banach algebras with trivial center. It is clear that such Banach algebras are not unital. Thank you.

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  • $\begingroup$ Presumably you want $A$ to be infinite-dimensional? $\endgroup$ – M.G. Aug 1 '18 at 11:55
  • $\begingroup$ I think this question is too broad, and should not be so open-ended. You should think harder about what particular properties you want your examples to have: what kinds of Banach algebras are you studying? $\endgroup$ – Yemon Choi Aug 1 '18 at 12:01
  • $\begingroup$ Nevertheless I will add another example to the list, but I strongly encourage you to make the question more limited. $\endgroup$ – Yemon Choi Aug 1 '18 at 12:04
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    $\begingroup$ It took me a second to realize that by a trivial center the OP means $Z(A)=\emptyset$ rather than $Z(A)=$ the underlying field. It is clear from the last sentence, but nevertheless it would be better stated explicitly. $\endgroup$ – M.G. Aug 1 '18 at 12:27
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    $\begingroup$ @M. G. By trivial I meant just the singleton $\{0\}$. That is, if an element commutes with all elements, then it is zero. $\endgroup$ – Fermat Aug 1 '18 at 19:02
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Take the ideal of compact operators on an infinite-dimensional normed space. Only zero commutes with everything.

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$\left\{ \begin{pmatrix} a & b \cr 0 & 0 \end{pmatrix} \colon a,b\in {\mathbb C} \right\}$ has trivial centre.

(This is the simplest non-trivial case of $\ell^1(B)$ where $B$ is a rectangular band, since I am guessing that you work in an academic scene where people have supposedly studied $\ell^1$-semigroup algebras. In particular one can easily generate infinite-dimensional examples in this way.)

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