The projection lemma (for example, Rudin's functional analysis book theorem 12.4) says that if $M$ is a closed vector subspace of a $\mathbb{C}$-Hilbert space $\mathcal{H}$ then $$\mathcal{H}=M\oplus M^{\perp}$$This lemma is a crucial ingredient in the proof of the Riesz representation theorem (stating that $\mathcal{H}\cong\mathcal{H}^{\ast}$), namely, in order to show surjectivity of the map $\mathcal{H}\to\mathcal{H}^{\ast}$.
For Hilbert C-star modules (for example defined in Blackadar) the Riesz representation theorem does not hold. If so far what I wrote is correct, I was wondering exactly how the projection lemma fails for that case?
Let $\mathscr{E}$ be an $A$-right-Hilbert module, where $A$ is a separable C-star algebra. Let $\mathscr{F}\subseteq\mathscr{E}$ be a closed (in the topology induced by the inner product) $A$-submodule. We would have liked to show that $$ \mathscr{E}=\mathscr{F}\oplus \mathscr{F}^{\perp} $$ and our goal is to see exactly why this fails for Hilbert $A$-modules.
Consider the Hilbert module $A := C([0,1])$ with its submodule $J = SA$, the suspension of $A$. Then $J^\perp = \{0\}$ and it follows that $J \oplus J^\perp \neq A$.