Banach algebras with trivial center

Question

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.

Answer 1

Take the ideal of compact operators on an infinite-dimensional normed space. Only zero commutes with everything.

Answer 2

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