I would like to understand the proof of Lemma 1, page 339 in [this](http://www.alainconnes.org/docs/book94bigpdf.pdf) book. Very briefly, the context is as follows: we have even dimensional oriented conformal manifold with the Hodge star operator chosen and we would like to construct a Fredholm module based on this data. In order to do so, we need an auxiliary operator, denoted by $S$ which has the property that the graph of $S$ coincides with the image of exterior derivative. There are several points which are not clear for me in the argument: 



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> Q1. Why any differential form in $\mathcal{H}\_0^+$ which is orthogonal to harmonic forms may be written as $\frac{1+\gamma}{2} d\alpha$ for some $\alpha$? 
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Hodge theory ensures us that such $\omega$ lies in the image of $d+d^\*$ and the claim should somehow follow from the fact that $d^\*=-\*d\*$ but I don't see how since there is no obvious commutation relation between the grading $\gamma$ and $d$. 



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> Q2. I don't see why $\|\frac{1+\gamma}{2} d \alpha \|\_2 =\| \frac{1-\gamma}{2}d\alpha \|\_2$. 
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In fact after simple computation this is equivalent to the fact that $\langle d \alpha, \gamma d\alpha \rangle = -\langle \gamma d \alpha, d \alpha \rangle$. But how to get this minus sign? I thought that $\gamma$ should be self adjoint. 



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> Q3. Even if we have this equality, does it really imply that $d\alpha$ is determined uniquely? 
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Leaving this specific context for a moment, we can imagine two vectors in $\mathbb{R}^2$ say $\xi\_1=(1,1)$ and $\xi\_2=(-1,1)$ and a projection $P(x,y):=(0,y)$ Then $P\xi\_1=P\xi\_2$ and $\| P \xi\_1 \|= \| (I-P)\xi\_1 \|$ but $(I-P)\xi\_1 \neq (I-P)\xi\_2$. In our context the role of $P$ is played by $\frac{1-\gamma}{2}$ (which indeed is a projection). 


I suspect that this uniqueness is needed in order to define $S$ in such a way that $S (\frac{1-\gamma}{2} d\alpha):=\frac{1+\gamma}{2} d \alpha$. But even so I still don't see why the graph of $S$ should be equal to the image of $d$ (image or closure of image?) since I can't get rid of the harmonic part in the expression $\omega+S \omega$ for $\omega \in \mathcal{H}\_0^{-}$ (which is mapped to 0 by $S$). 


I realize that these are technical questions and can be asked separetly, nevertheless I think that the whole context may be here important therefore I decded to ask all my questions in one post-I hope that this will be fine. 



Hodge theory, conformal manifolds and Fredholm modules-understanding the proof of one Lemma
Here is the answer to the questions. (Recall $V$ is a closed oriented conformal manifold of dimension $2n$). According to the Hodge-deRham decomposition, any $\omega\in L^2(V,\Lambda^nT^\*V) $can be written 
 $$\omega=h+d\alpha+d^\*\beta$$ where $h\in L^2(V,\Lambda^nT^\*V)$ is harmonic, $\alpha\in W^{1,2}(V,\Lambda^{n-1}T^\*V)$ and $\beta\in W^{1,2}(V,\Lambda^{n+1}T^\*V)$, moreover $\alpha$ (resp. $\beta$) is unique provided it is weakly coexact (resp. weakly exact). In fact, there is a unique $\eta\in W^{2,2}(V,\Lambda^{n}T^\*V)$, orthogonal to the space of harmonic form such that:
 $$\omega=h+(dd^\*+d^\*d)\eta.$$
 Now if $\gamma\omega=\omega$ and $\omega$ is orthogonal to the space of harmonic form then 
 $$\omega=d\alpha+d^\*\beta \ \mathrm{and}\ \omega=\gamma d\alpha+\gamma d^\*\beta=d^\*\tilde\alpha+d\tilde \beta$$
 where (for some $\varpi,\varpi'\in \{1,i\}$) $\tilde\alpha=\pm\varpi \gamma\alpha$ and $\tilde\beta=\pm\varpi' \gamma\beta$, by uniqueness we get 
 $$d\alpha=d\tilde \beta=\gamma d^\*\beta \ \mathrm{and}\ d^\*\beta=\gamma d\alpha=d^\*\tilde\alpha$$ so that 
 $$\omega=(1+\gamma)d\alpha=\frac{1+\gamma}{2} d\alpha.$$


For the next question, we have
 $$\langle d\alpha,\gamma d\alpha\rangle\_{L^2} =\pm\varpi \langle d\alpha,\star d\alpha\rangle\_{L^2}=\pm\varpi\int\_V d\alpha\wedge d\alpha=\pm\varpi\int\_V d(\alpha\wedge d\alpha)=0.$$


