A question about reduced C*-algebra of discrete group

Question

There is a quotation below:

Let $\Gamma$ be a discrete group and $\Lambda\subset \Gamma$ be a subgroup. The right cosets give a direct sum decomposition $$l^{2}(\Gamma)\cong\bigoplus l^{2}(\Lambda g)$$ and hence the left regular representation of $\Gamma$, when restricted to $\Lambda$, is a multiple of the left regular representation of $\Lambda$ (multiplicity equals the number of cosets). This implies

Proposition 2.5.9. If $\Lambda\subset\Gamma$ is a subgroup, then $C^{*}_{\lambda}(\Lambda)\subset C^{*}_{\lambda}(\Gamma)$ canonicallly. (The $C^{*}_{\lambda}(.)$ denotes the reduced C*-algebra of (.)).

I have three question about this quotation:

1. How to verify the direct sum decomposition $l^{2}(\Gamma)\cong\bigoplus l^{2}(\Lambda g)$ ?

2. How to comprehend that the left regular representation of $\Gamma$ is a multiple of the left regular representation of $\Lambda$?

3. How to get the Proposition 2.5.9.from the analysis?

Recall The reduced group C*-algebra of $\Gamma$, denoted $C_{\lambda}^{*}(\Gamma)$, is the completion of $\mathbb{C}(\Gamma)$ with respect to the norm $$||x||_{r}=||\lambda(x)||_{\mathbb{B}(l^{2}(\Gamma))}.$$

Answer 1

1. They are just saying that if $A=\bigcup_nA_n$, then $\ell^2(A)\simeq\bigoplus \ell^2(A_n)$. The direct sum has to be understood as a direct sum of Hilbert spaces, i.e. an $\ell^2$-sum.

2. The left regular representation of $\Lambda$ leaves each direct summand invariant. So we can think of it acting that way on $\ell^2(\Gamma)$ (as a block-diagonal operator).

3. By 1 and 2 one gets how to see $\lambda_\Lambda$ as $\lambda_\Gamma|_\Lambda$. So $C^*_\lambda(\Lambda)$ is generated by a subalgebra of $C^*_\lambda(\Gamma)$.