How to prove Character space of $M_n$(C) is empty by following question?

Question

Prove that Mn(C) is spanned by {AB−BA : A, B ∈ Mn(C)}. Deduce that Ω ( Mn(C) ) is empty, ie character space/Spectrum is empty.

Answer 1

For the second part of your question (independent from the first part): the characters here are $\star$-homomorphisms from $M_n(\mathbb{C})$ to $\mathbb{C}$. The kernel of this homomorphism must be a two-sided ideal. $M_n(\mathbb{C})$ has no nontrivial two-sided ideals, so the kernel can only be either the full algebra or zero. However, the range being at most one-dimensional, by rank nullity, kernel must have dimension at least $n^2-1$. So, the only choice is the zero homomorphism, which is not a character, which means, the character space is empty.

Regarding obtaining the second part from the first part: this follows directly from the definition of characters ($\star$-homomorphisms), since the range $\mathbb{C}$ is commutative.