Let $(W,S)$ be a Coxeter system. Let $q=(q_s)_{s\in S} \in \mathbb{R}^{\text{#}S}$ be a tuple of positive real numbers with $q_s=q_t$ whenever $s$ and $t$ are conjugate to each other. Follwing Davis', Proposition 19.1.1 we can build a (unique) $\ast$-algebra $\mathbb{C}_qW$ over $\mathbb{C}$ freely generated by basis elements $(e_w)_{w\in W}$ such that

\begin{equation}
e_se_w=
\begin{cases}
e_{sw} & \text{, if } \left|sw\right|>\left|w\right|\\
q_{s}e_{sw}+\left(q_{s}-1\right)e_w & \text{, if } \left|sw\right|<\left|w\right|
\end{cases}
\text{,}
\end{equation}
\begin{equation}
\nonumber
\left(e_w\right)^{\ast}=e_{w^{-1}}
\end{equation}
for $s\in S,w\in W$. This is the *multi-parameter (Iwahori-) Hecke algebra* associated to $q$.

Several sources suggest that, if the Coxeter group $W$ is finite, the Hecke algebra $\mathbb{C}_qW$ is always (abstractly) isomorphic to the group algebra. But I can not find a good reference for that statement. Can you give a reliable reference?

I'm also wondering if the mentioned isomorphism preserves the $\ast$-structure of $\mathbb{C}_q W$ (i.e. if it is an isomorphism of $\ast$-algebras or only an isomorphism of algebras).