Let $A= C([0,1])$ and $J= \{f \in A: f(0) = 0\}$. Consider the Hilbert $C^*$-module $E:= A \oplus J$ (with the obvious right $A$-action and inner product). I want to prove that $$q: E \to E: (f,g) \mapsto (f-g, 0)$$ is not adjointable. This is claimed in Lance's book on Hilbert $C^*$-modules, p22.

Here is what I tried. Assume to the contrary that $q$ is adjointable. Then there is $q^*: E \to E$ such that $$ (\overline{f-g})s= \langle q(f,g) , (s,t)\rangle = \langle (f,g), q^*(s,t)\rangle.$$

In particular, $q^*(s,t)$ does not depend on $t$ so we have $q^*(s,t) = q^*(s,0)$. Then I'm stuck.