$C^*$ algebras and differential geometry

Question

If $(A, G, \alpha)$ is a $C^*$-dynamical system where $G$ is a Lie group and $\alpha: G \rightarrow \operatorname{Aut}(A)$ is a continuous homomorphism (where $\operatorname{Aut}(A)$ is equipped with the topology of pointwise convergence), Connes defines $x \in A$ to be of class $C^\infty$ if the map $g \mapsto \alpha_g(x)$ from $G$ to $A$ is of class $C^\infty$ (in this paper: https://arxiv.org/pdf/hep-th/0101093.pdf). I am struggling how to understand how this map could be seen as $C^\infty$, as in my (limited) experience with smooth manifolds we need a smooth structure on the domain and codomain of a map, and while $G$ has a smooth structure, $A$ does not.

Answer 1

I'm definitely not an expert, but here is the very little I know. To get a derivative, what you need from the codomain is to take a difference, divide by a real number, and take a limit; so any topological vector space would work. Things are even clearer in a normed space, which a C$^*$-algebra is. On the domain side, you need to be able to consider small displacements and make them go to zero. There is an additional issue, which is that the domain is not commutative, so you will need to choose how to apply the displacement.

With the above in mind, this is the way it is (often? sometimes?) done, in the case you consider. Let $L(G)$ be the set of one parameter groups on $G$. Fix $X\in L(G)$. Then define $$ (D_X\alpha)(g)=\lim_{t\to0}\frac{\alpha(gX(t))-\alpha(g)}{t} $$ if the limit exists. Now $\alpha$ is said to be differentiable if the function $D\alpha:L(G)\times G\to A$ given by $$(D\alpha)(X,g)=D_X\alpha(g)$$ is well-defined and continuous. Iterating, one says that $\alpha\in C^n$ if the function $D^n\alpha:L(G)\times\cdots\times L(G)\times G\to A$ given by $$ D^n\alpha(X_1,\ldots,X_n,g)=(D_{X_1}D_{X_2}\cdots D_{X_n}\alpha)(g) $$ is well-defined and continuous.

Finally, $\alpha\in C^\infty$ if $\alpha\in\bigcap_nC^n$.

I took the above definitions from On Differentiability of Vectors in Lie Group Representations by Beltiţă and Beltiţă.