# Banach algebras with trivial center

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.

• Presumably you want $A$ to be infinite-dimensional? – M.G. Aug 1 '18 at 11:55
• 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? – Yemon Choi Aug 1 '18 at 12:01
• Nevertheless I will add another example to the list, but I strongly encourage you to make the question more limited. – Yemon Choi Aug 1 '18 at 12:04
• 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. – M.G. Aug 1 '18 at 12:27
• @M. G. By trivial I meant just the singleton $\{0\}$. That is, if an element commutes with all elements, then it is zero. – Fermat Aug 1 '18 at 19:02

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