spectrum of $C^*$ algebras

Question

When $A=\bigoplus B(\Bbb C^n)$ ($c_0$ direct sum),how to compute the spectrum of $A$ ?What about the conclusion If we replace the $\ell ^\infty $ direct sum with $c_0$ direct sum?

Answer 1

Let $\tau$ be a character of $A$. The composition $\tau\circ\pi_n$ where $\pi_n$ is the projection onto the $n^{\rm th}$ coordinate, is a character of $M_n(\mathbb C)$; which has none unless $n=1$.

So the only character of $A$ is $a\longmapsto a_1$, assuming that the first coordinate is one-dimensional.