I have a question in Haagerup's paper "An Example of a Non-Nuclear C*-algebra, which has the Metric Approximation Property", namely in Lemma 1.1.
Let $\Gamma$ a discrete (countable) group and $\varphi$ a positive definite function on $\Gamma$, the goal is to show that exists a unique, completely positive operator $M_\varphi: C_\lambda^* (\Gamma) \xrightarrow{} C_\lambda^* (\Gamma)$ such that for all $x \in \Gamma$, $$ M_\varphi \lambda_x = \varphi(x)\lambda_x. $$ The proof goes as follows: Let $(\pi,H,\xi)$ be the GNS triple associated to $\varphi$ and fix a Hilbert basis $\{e_n: n \in \mathbb N\}$. For every $n$, define the function $a_n: \Gamma \xrightarrow{} \mathbb C$ by $$ a_n(x) = \langle \pi(x)e_n , \xi \rangle. $$ We can prove that for all $x \in \Gamma$, $$ \sum\limits_{n \in \mathbb N} |a_n(x)|^2 = \varphi(e). $$ Moreover, it is easily seen through the Cauchy-Schwarz inequality that $a_n \in \ell^\infty(\Gamma)$, thus viewing that as multiplication operators acting on $\ell^2(\Gamma)$, define for $f \in C_\lambda^* (\Gamma) \subseteq B(\ell^2(\Gamma))$, $$ M_\varphi f = \sum\limits_{n \in \mathbb N} a_n f a_n^*. $$ My question is why is this operator well defined? I have tried applying to an element $b \in \ell^2(\Gamma)$ and use inequalities like the Minkowski integral one and fubini but always arrive at a divergent series.
In the paper Haagerup just says "because the operator $\sum\limits_{n \in \mathbb N} a_n a_n^* = \varphi(e)\cdot$ is bounded".