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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!

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up vote 2 down vote accepted

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.

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I see, the second fact you mentioned is also used to show $||.||_{max}$ is the largest norm on algebraic tensor of two C star algebras, thanks a lot! – ougao Jul 27 '14 at 22:59

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