Why is the natural map from maximal to reduced C star algebra surjective?

Question

In the book "Kazhdan' property (T)" the third book in this link, page 438, One sentence is "the regular representation defines a surjective *-homomorphism $\lambda_G: C^{*}(G)\to C^{*}_\text{red}(G)$".

I understand the existence of such homomorphism, but I do not understand why it is onto, could anyone help explain it or provide any reference?

Thanks in advance!

Answer 1

It is onto because the image contains a generating set for $C^*_{\text{red}}(G)$, and any C$^*$-algebra homomorphism maps a C$^*$-algebra onto a C$^*$-algebra.