Show that a specific linear map from a Hilbert module into a C*-algebra satisfies a certain equation

Question

Let $A$ be a C*-algebra, $\mathcal{H}$ be a Hilbert $A$-module and $B$ be a C*-algebra. Let $\pi: A \to B$ be a $*$-homomorphism and $\tau: \mathcal{H}\to B$ be a linear map such that $\tau(\xi)^*\tau(\eta)=\pi((\xi|\eta))$ for every $\xi,\eta\in\mathcal{H}$.

(This is from proposition 4.6.3 in C*-Algebras and Finite-Dimensional Approximations by Brown and Ozawa.)

Then it is "not hard to see" that $\tau(\xi a)=\tau(\xi)\pi(a)$ for every $\xi\in\mathcal{H}$ and $a\in A$. But how do you see that?

I tried the following:

$\tau(\xi)^*\tau(\eta a)=\pi((\xi|\eta a))=\pi((\xi|\eta)a)\pi((\xi|\eta))\pi(a)=\tau(\xi)^*\tau(\eta)\pi(a)$,

but then I cannot get rid of the $\tau(\xi)^*$, so how should on go one to solve this?

Thank you for any inputs.

Answer 1

We have \begin{align*} \|\tau(\xi a)-\tau(\xi)\pi(a)\|^2&=\|(\tau(\xi a)^*-\pi(a^*)\tau(\xi)^*)(\tau(\xi a)-\tau(\xi)\pi(a))\|\\ &=\|\tau(\xi a)^*\tau(\xi a)-\tau(\xi a)^*\tau(\xi)\pi(a)-\pi(a^*)\tau(\xi)^*\tau(\xi a)+\pi(a^*)\tau(\xi)^*\tau(\xi)\pi(a)\|\\ &=\|\pi((\xi a|\xi a))-\pi((\xi a|\xi))\pi(a)-\pi(a^*)\pi((\xi|\xi a))+\pi(a^*)\pi((\xi|\xi))\pi(a)\|\\ &=\|\pi(a^*(\xi|\xi)a)-\pi(a^*(\xi|\xi)a)-\pi(a^*(\xi|\xi)a)+\pi(a^*(\xi|\xi)a)\|\\ &=0. \end{align*}