Let $\mathcal{A}$ be a *-algebra and $p\in M_N(\mathcal{A})$ an orthogonal projection.
I need to show that $\nabla=p\circ d$ defines a connection on $\mathcal{E}=p\mathcal{A}^N$, where $d$ is acting on each copy of $\mathcal{A}$ as the commutator $[D,\cdot]$. Hence, I need to show that the Leibniz rule holds. In this case
$(p\circ d)(\eta a)= (p\circ d)(\eta)a+\eta\otimes_\mathcal{A} da \quad$ with $\eta\in p\mathcal{A}^N$ and $a\in\mathcal{A}$
It holds that $\eta_i=\sum_{j=1}^N p_{ij}a_j$. Therefore it follws that $(p\circ d) (\eta a) = p ([D,\sum_j p_{1j}a_ja],\dots, [D,\sum_j p_{Nj}a_ja])^T $
The k-th component $((p\circ d) (\eta a))_k$ has then the form
How do I obtain $\eta\otimes_\mathcal{A} da$ instead of $\eta da$ ?
Thanks for your help