center of finite dimensional $C^*$ algebra

Question

If $A$ is a finite dimensional $C^*$ algebra,is the center $\mathcal{Z}(A)$ of form $\lambda 1_{A}$,where $\lambda\in \mathbb{C}$?

Answer 1

A finite-dimensional C$^*$-algebra is of the form $$A=\bigoplus_{j=1}^n M_{k(j)}(\mathbb C).$$ The centre consists precisely of the operators of the form $$ T=\bigoplus_{j=1}^n \lambda_j\,I_{k(j)}. $$ It follows that $Z(A)=\mathbb C^n$.