Multiplier algebra of the group $C^*$-algebra for a discrete group

Question

Let $\Gamma$ be a discrete group, and let $\mathbb{C}\Gamma$ be its group algebra, so that an element of $\mathbb{C}\Gamma$ is a finite formal linear combination $$\sum_{i=1}^k c_{\gamma_i}\gamma_i,$$ where $c_{\gamma_i}\in\mathbb{C}$ and $\gamma_i\in\Gamma$. Multiplication is induced by multiplication in $\Gamma$, extended linearly.

Then $\mathbb{C}\Gamma$ can be viewed as a $*$-subalgebra of $\mathcal{B}(l^2(\Gamma))$ in a natural way via the left-regular representation. Let $C^*_r(\Gamma)$ denote the completion of $\mathbb{C}\Gamma$ in the operator norm. Then $C^*_r(\Gamma)$ is a unital $C^*$-algebra called the reduced group $C^*$-algebra.

Question: Is $C^*_r(\Gamma)$ its own multiplier algebra?

Answer 1

Note that for a $C^*$-algebra $A$ with multiplier $C^*$-algebra $M(A)$ we have that

$$A \mathrm{\ unital}\iff M(A) = A$$

To see this, note that if $A$ is unital then the identity of $M(A)$ lives in $A$ and since $A$ is an ideal in $M(A)$ we get $A = M(A)$. The converse is obvious.

Your specific case follows easily from this.