The representation π in (6.35) can be extended to a surjective map $$\mathbb{I}⊗π : CC(\mathcal{E}) → C(\mathcal{E})$$ , (8.13) namely, any compatible connection is the composition of π with a universal compatible connection.
The above is taken from hep-th/9701078v1 page 121. I would like to know what does the notation $CC({E})$ stand for. Anyone familiar with the verbiage of noncommutative geometry please have a crack at it?
In the absence of a Hermitian connection on a right $\mathcal{A}$-module $\mathcal{E}$, $CC(\mathcal{E})$ denotes the affine space of all universal connections on $\mathcal{E}$ (p. 114); when $\mathcal{E}$ is also given a Hermitian structure, then, by slight abuse of notation, $CC(\mathcal{E})$ also denotes the affine space of all universal connections on $\mathcal{E}$ compatible with the Hermitian structure (p. 117), and this is the definition that's relevant on p. 121. What (8.13) is saying is that every connection on a finitely generated projection right $\mathcal{A}$-module $\mathcal{E}$ with respect to the noncommutative de Rham calculus induced by the spectral triple $(\mathcal{A},H,D)$ can be obtained as the "representation" of a universal connection on $\mathcal{E}$ (equivalently, can be lifted to a universal connection).